Using the roots of the quadratic equation the equation is solved to be
f(x) = (x + 2)(x − 3)How to get the equation of the quadratic equationInformation from the problem include
The quadratic function f(x) has roots of −2 and 3 and point (2, −4) lies on f(x)
from the roots
x = -2 and x = 3
x + 2 = 0 and x - 3 = 0
(x + 2)(x - 3)
passing through points (2 , -4)
y = a(x + 2)(x - 3)
-4 = a(2 + 2)(2 - 3)
-4 = a * 4 * -1
-4 = -4a
a = 1
the quadratic equation is f(x) = (x + 2)(x − 3)
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help me solve this please
How do you prove a square is congruent?
Answer:
How To Find if Triangles are Congruent 1 SSS (side, side, side) 2 SAS (side, angle, side) 3 ASA (angle, side, angle) 4 AAS (angle, angle, side) 5 HL (hypotenuse, leg)
hope this helps
mark me brainliest
Step-by-step explanation:
suppose that 10 golfers enter a tournament and that their respective skill levels are approximately the same. what is the probability that one of the first three golfers that registered for the tournament will win?
The correct answer c. 0.300
All the golfers have similar skill levels, which means that they are equally as likely to win the tournament.
The sum of all 10 golfers probability to win the tournament is 100/10 = 10%.
What is the probability that one of the first three golfers that registered for the tournament will win?
Each one of them has a 10% probability of winning
3*10% = 30% = 0.3
So the correct answer is:
c. 0.300
Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.
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Full Question :Suppose that 10 golfers enter a tournament and that their respective skill levels are approximately the same. What is the probability that one of the first three golfers that registered for the tournament will win?
a. 0.111
b. 0.001
c. 0.300
d. 0.9
e. 0.850
How do you find the multiplicity of roots?
The multiplicity of a root of a polynomial can be determined by the factorization of the polynomial.
If a root is repeated multiple times, then this root is said to have a higher multiplicity. For example, if a polynomial has the form a(x-r)^2, then the multiplicity of root r is 2. This is because the polynomial can be written as a(x-r)(x-r), and the root r is repeated twice.
To calculate the multiplicity of a root, the polynomial must be factored into its linear factors. The multiplicity of each root is then equal to the power of the corresponding linear factor. For example, if a polynomial is factored into (x-r)^3(x-s), the multiplicity of root r is 3 and the multiplicity of root s is 1.
In general, if a polynomial is written in the form [tex]a*(x-r1)^k1*(x-r2)^k2*...*(x-rn)^kn[/tex], then the multiplicity of root ri is ki. Therefore, the multiplicity of the roots can be calculated by factoring the polynomial and finding the power of the linear factors.
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3) Find the percentage change. State whether the change is an increase or decrease.
a)
A
Before: 34 shirts
Now: 62 shirts
b)
Original: 36.79
New: 22.88
Answer:
(a) The percentage change is 82.35 . The change is decreasing.
(b) The percentage change is -37.80. The change is decreasing.
Step-by-step explanation:
(a) percentage change- the change between two values in percent is called a percentage change .
formula - final value-initial value /initial value *100
here , 62-34/34*100
since , the value is positive , the value is increasing .
(b) percentage change- the change between two values in percent is called a percentage change .
formula - final value-initial value /initial value *100
here , 22.88-36.79/36.79*100
since , the value is negative , the value is decreasing.
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Answer:
a) 82% Increase
b) -37% decrease
Step-by-step explanation:
a)
Percentage :-
= ((62-34) / 34) * 100
= 82.3529 ≈ 82% ------> increase
b)
Percentage:
= ((22.88-36.79) / 36.79) * 100
= -37.8 ≈ -37% ------> decreased
I hope my answer helps you
whoever answer first gets brainlist
Answer:
$165
Step-by-step explanation:
5% interest rate = 1 month
10% interest rate = 2month
10% = 0.1
Take $150 times 0.1 = $15
Find the total cost by taking
$150 + $15 = $165
So, the total cost of Jill's uniform is $165
What are the top 3 factors that contribute to your success?
The top three factors that contribute leading to my success are
HonestyPurpose and Intuition.There are several factors that I believe are important to a successful venture, business or individual. But in my opinion my main factors are:
Honesty really is the best policy. If you aren't honest with your work and yourself, you won't be successful.Purpose is born from stronge self-honesty . You can find a lot of freedom working for yourself, but make no mistake, it's work. Life is meaningless without any purpose of life.Getting started on your path to success isn't always obvious. Through experience, I have learned that sometimes you need to try different states of mind and learn along the way.To learn more about sucess factors , refer:
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Which of these is, or describes, a geometric sequence with a common ratio of 2? select two answers that are correct.
A geometric sequence with a common ratio of 2 is E . F( n ) = 5 * 2^n where n = 1 , 2 , 3 .
Given :
A. -5 , -3 , -1 , 1 , 3
common ratio r = -3/-5
it is not the option
B . 0.0625 , 0.125 , 0.25 , 0.5 , 1
Common ratio :
r = 0.125 / 0.0625
= 0.5
it is not the option .
C. a[tex]_n[/tex] = 1/2 * [tex]a_{n-1}[/tex] where a1 = 2
a2 = 1/2 * a1
= 2/2
= 1
common ratio = 1/2
it is not the option
D. a[tex]_n[/tex] = [tex]a_{n-1}[/tex] + 2 where a1 = 3
a2 = a1 + 2
= 3 + 2
= 5
common ratio = 5/3
it is not the answer
E. F( n ) = 5 * 2^n where n = 1 , 2 , 3
a1 = 5 * 2^1
= 10
a2 = 5 * 2^2
= 20
common ratio = 20 / 10
r = 2
It is the correct option.
F. f ( n ) = n^2 where n = 1 , 2 , 3
a1 = 1^2
a2 = 2^2
= 4
common ratio = 4/1 = 4
it is not the correct option.
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Full question :
Which of these is, or describes, a geometric sequence with a common ratio of 2? select two answers that are correct.
A. -5 , -3 , -1 , 1 , 3
B . 0.0625 , 0.125 , 0.25 , 0.5 , 1
C. a[tex]_n[/tex] = 1/2 * [tex]a_{n-1}[/tex] where a1 = 2
D. a[tex]_n[/tex] = [tex]a_{n-1}[/tex] + 2 where a1 = 3
E. F( n ) = 5 * 2^n where n = 1 , 2 , 3
F. f ( n ) = n^2 where n = 1 , 2 , 3
x^2 = 225
Explain the first step in solving this equation.
Solve and check your solution.
Answer:
Step-by-step explanation:
x^2 = 225
x = √15
How do you know if the graph opens upward or downward?
The graph of the quadratic equation which represents the parabola can be opened upward or opens downward depending upon the nature of the equation and coefficient
If the equation is in the form of [tex]y^{2} =ax+b[/tex] where [tex]a[/tex] is called the leading coefficient
The graph of a quadratic function opens upward or downward depending on the value of the leading coefficient a if the leading coefficient is greater than zero, ([tex]a[/tex]>0) the parabola opens upward, and if the leading coefficient is less than zero,([tex]a[/tex]<0) the parabola opens downward.
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Daphne participated in a challenge for National November Writing Month. All of the stories she
wrote for the challenge were more than 4 pages long.
Let z be the number of pages in a story. Which inequalities represent lengths of stories
more than 4 pages long? Select ALL that apply.
1: X <4
2: x >4
3: 4> x
4: 4< x
None of the given option is correct as for relating z i.e number of pages and pages wrote by Daphne we need a variable z in options.
What are inequalities?In mathematics, "inequality" refers to a relationship between two expressions or values that is not equal to each other. Therefore, inequality emerges from a lack of balance. When two quantities are equal, we use the sign "=," and when they are not equal, we use the symbol "," which stands for "not equal." The first value may be greater than (>), less than (<), greater than equal to (>=), or less than equal to (<=) the second value if the two values are not equal.
The words "inequality" and "inequalitas," which mean "unequal, unlike, distinct," and "changeable," respectively, derive from the Old French and Latin, respectively. The magnitude, number, and intensity of two quantities or expressions are compared using this phrase.
For given situation correct relation will be z>4 which is not given in any of the option.
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What is the quotient 6x4 15x3 10x2 10x 4 3x2 2 )? 6 points group of answer choices 2x2 − 5x 2 2x2 5x − 2 2x2 − 5x − 2 2x2 5x 2?
The quotient is 2x2+5x2+2.
The rational root theorem, also known as the rational root test, is an algebraic theorem that states that for a polynomial equation in one variable with integer coefficients to have a solution (root) that is a rational number, both the constant term (the term without a variable) and the leading coefficient (the coefficient of the highest power) must be divisible by the fraction's denominator.
Since we have already provided it
[tex]\frac{(6x^2+15x^3+10x^2+10x+4)}{(3x^2+2)}[/tex]
We will now factor the numerator to determine the quotient:
Use the theorem that says [tex]a_0=4[/tex] and [tex]a_n=6[/tex].
[tex]a_0[/tex] divisions are 1, 2, and 4.
[tex]a_n[/tex] is divided into 1, 2, 3, and 6.
Check the following logical figures as a result: ±[tex]\frac{1,2,4}{1,2,3,6}[/tex]
[tex]-\frac{2}{1}[/tex] is a root of the expression, so factor out [tex]x+2[/tex]
[tex]=(x+2)\frac{6x^4+15x^3+10x^2+10x+4}{x+2}[/tex]
[tex]= \frac{6x^4+15x^3+10x^2+10x+4}{x+2} \\=6x^3+3x^2+4x+2[/tex]
We'll factor it in once more now:
[tex]=(6x^3+3x^2)+(4x+2)\\=2(2x+1)+3x^2(2x+1)\\=(2x+1)(3x^2+2)[/tex]
Finally, we have our factored form:
[tex]= (x+2)(2x+1)(3x^2+2)\\=\frac{(x+2)(2x+1)(3x^2+2)}{3x^2+2} \\=(x+2)(2x+1)\\=2x^2+5x+2[/tex]
The complete question is:-
What is the quotient: (6x4 + 15x3 + 10x2 + 10x + 4) ÷ (3x2 + 2)?
a. 2x2 – 5x + 2
b. 2x2 + 5x – 2
c. 2x2 + 5x + 2
d. 2x2 – 5x – 2
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Rays are shown from point P, the center of dilation, through the vertices of angles QRS. Dilate angles QRS by a scale factor of 2
The dilation of the angle with vertices QRS has been dilated by scale factor 2.
What is the transformation of a graph?Transformation is rearranging a graph by a given rule it could be either increment of coordinate or decrement or reflection.
If we reflect any graph about y = x then the coordinate will interchange it that (x,y) → (y,x).
As per the given angle QRS.
The length of QS = 2 grids and RS = 3 grids.
By dilated by 2 scale factor.
The length QS must become 2 x 2 = 4 grids and RS by 3 x 2 = 6 grids.
The R and Q must be in dotted lines as shown below.
Q'R'S' is the dilated figure.
Hence "The angle with vertex QRS has been enlarged by a scale factor of 2".
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How do I write an algebraic expression?
An equation is like a sentence, just using a different language. Mathematical equations are telling you to do something, giving you information on how to respond, or come up with an answer. It's up to you to learn this language. However, once you know it, you can write in this language.
A mathematical equation, also known as a statement, shows that two separate expressions are equal. The two expressions are usually connected by the "equal-to" (=) sign. You can find the expressions in a mathematical statement by looking at the left-hand side and the right-hand side of the equation, using the = sign as the point of reference.
The most popular equations are algebraic equations or expressions, in which letters are used as variables. Variables are placeholders that can assume any value. They can also be referred to as unknowns. You must solve for the correct value of the variable to make the equation true.
Let us look at an example to understand.
4x + 2x = 12
The letter x is the variable. You must find the value of x. Finding this value will make the right-hand side of the equation equal to the left-hand side. Let's solve this together.
In this case, since both variables are the same, you can add the values on the left-hand side.
6x = 12
Now, we divide both sides by the value of '6', which will help us solve for x. Consider this to be a way to unravel the equation. Also, remember that whatever is done to one side must be done to the other.
x = 2
This is exactly how to write an algebraic expression.
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What is an equivalent fraction to 7 12?
The equivalent fraction of 7/12 is 14/24
The given fraction is 7/12
The given fraction is in the form of simple fraction. Simple fraction consist of the numerator and the denominator. The term on the top is called numerator and the term on bottom is called denominator.
The fraction is 7/12
The equivalent fractions means that the value of the both fraction will be equal
To find the equivalent fraction of 7/12 multiply both numerator and the denominator of the simple fraction by the same digit
Here multiply both numerator and denominator by 2
(7×2) / (12 × 2) = 14 / 24
Therefore, the equivalent fraction is 14/24
I have answered the question in general, as the given question is incomplete
The complete question is
What is an equivalent fraction to 7/12?
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24 is 80% of what number? 26 28 30 48
Answer:
26
Step-by-step explanation:
ehtan put $700 into a certificate of deposit(cd) account that earns 1.8% interest each year. What will the interest of the cd account be after 6 years
The interest on Ethan's CD account will be $700 * 1.8% = 12.6$ dollars after one year. Since the CD earns 1.8% interest each year, the interest will be the same amount each year, so the total interest on the CD after 6 years will be $12.6 * 6 = 75.6$ dollars. This means that the CD will be worth $700 + 75.6 = $775.6$ after 6 years.
What is the value of 5 2 10 4 2 5x5 6?
The simplification of the given expression is found as 2.
Explain the concept of the indices?We can determine how several times a phrase has already been multiplied in and of itself using an index, which is a tiny number.Indexes is the plural of index.The stated expression is-
= (5⁻² x 10⁻⁴) / (2⁻⁵ x 5⁻⁶)
Taking the reciprocal of the numbers.
= (2⁵ x 5⁶) / (5² x 10⁴)
Cancel the common power
= (2⁵ x 5⁴) / ( (2x5)⁴)
Expanding the power
= (2⁵ x 5⁴) / ( 2⁴ x 5⁴)
Again cancel the common power.
= 2
Thus, the simplification of the given expression is found as 2.
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The correct question is-
What is the value of (5⁻² x 10⁻⁴) / (2⁻⁵ x 5⁻⁶)?
What is the solution Ofsimultaneous equations 5x 3y 8 and 3x Y 2?
The solution of the simu1taneous equation 5x−3y=8; 3x+y=2 is x=1andy=−1.
What is simultaneous equations?Two or more algebraic equations that share variables, such as x and y, are said to be simultaneous equations. Since the equations are solved simultaneously, they are known as simultaneous equations. These equations alone could have an endless number of solutions.A finite set of equations for which common solutions are sought is referred to in mathematics as a set of simultaneous equations, often known as a system of equations or an equation system. In order to create a single equation that contains the other unknown, one of the unknowns must be removed or eliminated. It is best to use examples to explain the process. Example Address the parallel equations. 3x + 2y = 36 (1) 5x + 4y = 64 (2) . y = 6 So, x = 8 and y = 6 is the complete solution.Consider the equation 3x+y=2
Multiply the equation by 3 so that we can eliminate y
3(3x+y=2)=3×2
9x+3y=6
Adding it with 5x−3y=8
9x+3y+5x−3y=6+8
14x=14
x=1
Putting the value of x in equation 3x+y=2
3×1+y=2
3+y=2
y=2−3
y=−1
So, we have x=1andy=−1
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A company has 95,000 online followers and gains 100 new followers every day. Another
company starts with 0 online followers and gains 350 new followers every day.
After how many days will the two companies have the same number of followers?
The number of days when the two companies will have the same number of followers is 380 days.
How to calculate the number of days?The first step is to form linear functions that represent the information of the two companies. The form of the functions is:
Total followers = initial number of followers + (number of new followers x number of days)
The linear function that represents the online followers for the first company is:
Total followers = 95,000 + 100d
The linear function that represents the online followers for the second company is: Total followers = 350d
When the companies have the same number of followers the two above equations would be equal 95,000 + 100d = 350d
95,000 = 350d - 100d
95,000 = 250d
d = 95,000 / 250
d = 380
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What is the inverse of 8?
The additive inverse of the number 8 is " -8 " .
The Inverse (Additive) of the number is defined as the number that is added to a given number to make the sum zero ;
that means : If a number is added to its additive inverse, then the sum of both numbers becomes 0 .
So , that means to find the additive inverse of "8" , we have to find the number such that on adding it to 8 , we get the sum as 0 .
let the additive inverse of 8 be = x ;
So , x + 8 = 0 ;
x = -8 .
So , putting the value in above equation, 8 - 8 = 0 .
Therefore , the additive inverse is "-8" .
The given question is incomplete , the complete question is
What is the additive Inverse of 8 ?
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For what values of x and y are the triangles to the right congruent by HL?
For x = 5 and y = 3 the given triangles are congruent.
What is congruent triangles?
In any orientation, two triangles are said to be congruent if their three sides and three angles are equal.
Therefore, if two triangles have the same number of sides on each side, they are said to be congruent. Additionally, we can determine that they are congruent if we have a side, an angle between the sides, and then another side that is congruent, in that order: side, angle, side.
Let the given triangles are congruent.
So, their sides are equal.
⇒ x = y + 2 ..(1)
3y = x + 4 ..(2)
Plug value of x in equation (2)
3y = (y + 2) + 4
3y = y + 6
2y = 6
y = 3
Plug y = 3 in equation (1)
x = 3 + 2
x = 5
Hence, for x = 5 and y = 3 the given triangles are congruent.
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what is the answer in decimal form?
The standard deviation of the given population is 7.72.
What is population?
All of the components of the data set are included in the population, as are parameters like mean and standard deviation that can be used to measure the population. The term "population" denotes the complete group of individuals, things, events, etc.
Given data,
53, 36, 36, 44, 31
We know that,
Standard deviation of a population is given by :
= √ [ ∑ ( X - ц )²/N
Where, X = each data point
ц = mean of data of population
N = No. of data points in population = 5
We need mean,
So, mean of data of population = (53 + 36 +36 +44 + 31)/5
= 40.
So, By applying formula :
SD = √ [ ∑ ( X - ц )²/N ]
= √1 / 5 [ (40-53)² + (40-36)² + (40-36)² + (40-44)² + (40-31)²
= √ ( 169 + 16 + 16 + 16 + 81 )/5
= √298/5
=7.72.
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What is special about rigid transformations?
transformations and rigid body symmetric transformations. Geometric figures' sizes and shapes are unaltered throughout a stiff transformation (also known as an isometric transformation).
What type of transformation is rigid Why?A stiff transformation in mathematics, commonly referred to as a Euclidean transformation or a Euclidean isometric transformation, is a geometric transformation of Euclidean space that keeps the Euclidean distance between each pair of points. Transformations of rigid bodies can take the shape of rotations, translations, reflections, or any combination of these.
transformations and rigid body symmetric transformations. Geometric figures' sizes and shapes are unaltered throughout a stiff transformation (also known as an isometric transformation). a unique kind of metamorphosis when the characters' sizes or shapes are left unaltered.
Area = [tex]\sqrt{s(s - a)(s - b)(s - c) }[/tex]
and, here s = half the perimeter,
[tex](a + b + c)/2.[/tex]
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How to find duplicate elements in array in java using HashMap?
In this function, we use Java's HashMap to locate duplicate elements in an array. The input array's items are kept as keys and their occurrences as values in a hash map. Any key that has a value greater than one (>1) is considered to be a duplicate element.
In Java, you may discover duplicates in an array using a variety of ways. This article will teach you how to use Java's Brute Force method, Sorting method, HashSet, HashMap, and Java 8 Streams to locate duplicate elements in an array. Let's examine each one in turn.
Private static void new HashMap(); HashMapInteger, Integer> map = new
HashMap; findDuplicatesUsingHashMap(int[] inputArray);
for (int element: inputArray)
if (map.get(element) == null) map.put(element, 1);
else map.put(element, map.get(element)+1);
Set> Set entry "Integer, Integer" map entrySet =. entrySet();
for (EntryInteger, Integer> entry: entrySet)
if(entry.getValue() > 1)
System.out.println("Duplicate Element: "+entry.getKey()+" - found "+entry.getValue()+" times.");
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1/8n=−7 the value of n
Answer:
n = -56
Step-by-step explanation:
1/8n=−7
1/8 * n = −7
n * 1/8 = −7
n/8 = −7
8 * n/8 = -7 * 8 ==> multiply each side by 8 to isolate n and remove fractions
n = -56
__________________________________________________________
Hello! In this question, we are going to solve for the variable "n".
__________________________________________________________
Explanation:
In the question, we were given the equation:
[tex]\frac{1}{8} n=-7[/tex]
We will solve for the variable "n" by getting it by itself, which would then be our "n" value.
__________________________________________________________
Solve:
Let us solve for n:
[tex]\frac{1}{8} n=-7\\\\\text{To get "n" by itself, we will multiply by 8 to eliminate the fraction}\\\text{Whatever you do to the left side of the equal sign, you do to the right}\\\text{Multiply both sides by 8}\\\\8\times\frac{1}{8} n=-7\times8\\\\n=-56\\\\\text{Since "n" is now by itself, we can't simplify any further, therefore our }\\\text{ "n" value would be \boxed{-56}.}[/tex]
__________________________________________________________
Answer:
[tex]\boxed{n=-56}[/tex]
__________________________________________________________
Additional info:
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where do the asymptotes occur in
y = 1/3 cot 2x
how can i know?
someone help!
Answer:
[tex]\displaystyle{x=\dfrac{\pi}{2}n}[/tex] for n is any integers
Step-by-step explanation:
To know the asymptotes, first, we must know values of x that we turn y-value into an undefined value.
We know that:
[tex]\displaystyle{\cot x = \dfrac{1}{\tan x}}[/tex]
Now we have to find value of x that turns the identity above into undefined value, and that is [tex]\displaystyle{x=n\pi}[/tex] where n is any integers. (This gives 1/0 for all x = nπ)
Therefore, a function [tex]\displaystyle{\cot x}[/tex] has asymptote lines at [tex]\displaystyle{x=n \pi}[/tex] for n is integers.
If we consider the given problem:
[tex]\displaystyle{y=\dfrac{1}{3}\cot 2x}[/tex]
We have to find values of x that turn y-value undefined. We know that [tex]\displaystyle{x=n\pi}[/tex] is asymptotes for [tex]\displaystyle{\cot x}[/tex]. Therefore, [tex]\displaystyle{2x=n\pi}[/tex] has to be asymptotes for [tex]\displaystyle{y=\dfrac{1}{3}\cot 2x}[/tex].
Hence, the asymptotes occur at [tex]\displaystyle{x=\dfrac{\pi}{2}n}[/tex] by solving the equation and for n is any integers.
How many numbers less than 1000 are divisible by 3?
There are 333 numbers less than 1000 that are divisible by 3.
To find this, you can divide 1000 by 3 to get 333 with a remainder of 1. Since the numbers less than 1000 that are divisible by 3 do not include 1000, you subtract 1 from 333 to get 332. However, 0 is also divisible by 3, so you should add 1 to 332 to get 333 as the final answer.
In mathematics, what is a divisible?When a number divides evenly (i.e., leaves no residue), it is said to be divisible. For instance, since 2 can be divided by 34 equally, 34 may be divided by 2. 34 is not divisible by 3 since doing so would leave us with a remainder.
How are three divided?A whole number is said to be divisible by three according to the three-digit rule of division if the sum of all of its digits is precisely divided by three. Without actually doing division, it is possible to establish whether a number may be divided by three.
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what values for m and n that make this number sentence true: 1/2+9/10+m+n=2.
1/2+9/10+m+n=2 is true when m and n have values { (m, n) : m = 3/5 - n, ∀ n and m ∈ R }.
What is number?
An arithmetic value known as a number is used to calculate and represent a quantity.
Given, 1/2+9/10+m+n=2
m+ n = 2 - 1/2 - 9/10
m+ n = 3/5
m = 3/5 - n
So, If n= 0, m = 3/5
Hence, we can define as { (m, n) : m = 3/5 - n, ∀ n and m ∈ R }
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Find a set of vectors spanning the null space of A= 1 1 2 -1 2 3 6 -2 -2 1 2 2 0 -2 -4 0
The set of vectors spanning the null space of A is {(0,0,z)}.
A set of vectors spanning the null space of a matrix A is a set of vectors that, when multiplied by the matrix A, result in the zero vector. In other words, these vectors are in the null space of A because they are mapped to the zero vector under the transformation represented by A.
To find a set of vectors spanning the null space of matrix A, you can use the following steps:
Write the matrix A in row-reduced echelon form.Identify the pivot columns in the matrix. These are the columns that contain the leading 1's in the row-reduced echelon form.For each pivot column, assign a variable to the corresponding pivot row.For each pivot column, set the variables corresponding to all other rows in the column equal to 0.For each non-pivot column, set the variables corresponding to the rows in the column equal to a free variable.The set of vectors formed by the values of the variables is a set of vectors spanning the null space of A.To find a set of vectors spanning the null space of the matrix A= 1 1 2 -1 2 3 6 -2 -2 1 2 2 0 -2 -4 0:
Write the matrix A in row-reduced echelon form:
1 1 2 | -1
0 0 0 | 0
0 0 0 | 0
The pivot columns are the first and second columns, which contain the leading 1's.
Assign variables x and y to the pivot rows:
1 1 2 | -1
0 0 0 | 0 x
0 0 0 | 0 y
Set the variables corresponding to all other rows in the pivot columns equal to 0:
1 1 2 | -1
0 0 0 | 0 0
0 0 0 | 0 0
Set the variables corresponding to the rows in the non-pivot column equal to a free variable z:
1 1 2 | -1
0 0 0 | 0 0 z
0 0 0 | 0 0 z
The set of vectors spanning the null space of A is {(0,0,z)}.
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