[tex]\bf ~~~~~~\textit{parabola vertex form} \\\\ \begin{array}{llll} \stackrel{\textit{we'll use this one}}{y=a(x- h)^2+ k}\\\\ x=a(y- k)^2+ h \end{array} \qquad\qquad vertex~~(\stackrel{}{ h},\stackrel{}{ k}) \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ y=0.25(x+5)^2\implies y=0.25[x-(\stackrel{h}{-5})]^2+\stackrel{k}{0}~\hfill \stackrel{\textit{vertex}}{(-5,0)} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\bf \stackrel{\textit{the y-intercept occurs when x =0}~\hfill }{y=0.25(0+5)^2\implies y=0.25(5)^2}\implies y=6.25~\hfill \stackrel{\textit{y-intercept}}{(0, 6.25)}[/tex]
To identify the vertex and the y-intercept for the function \( y = 0.25(x + 5)^2 \), we will use our knowledge of quadratic functions and their properties.
Vertex:
The given function is in the form of \( y = a(x - h)^2 + k \), where (h, k) is the vertex of the parabola. This form is commonly referred to as the vertex form.
For the given function, \( y = 0.25(x + 5)^2 \), we can see that it is equivalent to \( y = 0.25(x - (-5))^2 + 0 \), indicating that the vertex (h, k) is at (-5, 0). This means the parabola opens upwards (because the coefficient of \( (x + 5)^2 \) is positive) and has its vertex point at x = -5 and y = 0.
Y-intercept:
The y-intercept of a function is found by evaluating the function when x = 0.
So we'll substitute x = 0 into the function to find the y-coordinate of the y-intercept:
\( y = 0.25(0 + 5)^2 \)
Now, we'll calculate the value inside the parentheses:
\( 0 + 5 = 5 \)
Then, we raise it to the power of 2:
\( 5^2 = 25 \)
Finally, we multiply by 0.25:
\( y = 0.25 \times 25 = 6.25 \)
Therefore, the y-intercept where the graph crosses the y-axis is at the point (0, 6.25), where the x-coordinate is 0 and the y-coordinate is the value we just calculated, 6.25.
In summary, the vertex of the function \( y = 0.25(x + 5)^2 \) is (-5, 0), and the y-intercept is at (0, 6.25).
How do you work out 15 divided by 25
Answer:
Step-by-step explanation:
15/25. Divide both sides by a common number which is 5 3/5 is the final answer.
The result of the given mathematical expression is [tex]\frac{3}{5}[/tex] or 0.6.
What is a mathematical expression?"A mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context."
Given mathematical expression is
= (15 ÷ 25)
[tex]= \frac{15}{25}\\= \frac{3}{5}\\= 0.6[/tex]
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What is the range of the exponential function shown below? F(x) = 11 • (1/3)x
Answer:
[tex](0, \infty)[/tex]
Step-by-step explanation:
By definition all the exponential functions of the form
[tex]f (x) = a (b) ^ x[/tex]
Where a is the main coefficient and b is the base they have range [tex](0, \infty)[/tex]
Whenever [tex]a> 0[/tex] and b> 0.
In this case the function is:
[tex]f (x) = 11(\frac{1}{3}) ^ x[/tex]
Note that for this function [tex]a = 11> 0[/tex] and [tex]b =\frac{1}{3}>0[/tex]
Therefor the range is: [tex](0, \infty)[/tex]
need help asap please
Answer:
y = -7
Step-by-step explanation:
The easisest way to find the slope of this line is to use slope-intercept form.
Slope-intercept form:
y = mx + b
Where m = slope and b = y -intercept
In this graph, the y-intercept is -7. However, the line doesn't have a slope since its a straight horizontal line.
So, the mx part of the equation isn't a part of this new equation.
So, your equation would just y = -7
which function is odd check all that apply
a. y=sin x
b. y=csc x
c. y=cot x
d. y=sec x
Answer:
a) y = sin x
b) y = csc x
c) y = cot x
Step-by-step explanation:
only d is even
how do I solve y^4-13y^2+36=0
Answer:
The roots are {-3, -2, 2, 3}.
Step-by-step explanation:
The trick here is to represent y² by some other letter, such as x. If we do that, then y^4-13y^2+36=0 becomes x² - 13x + 36 = 0.
Recognize that the factors -4 and -9 of 36 sum up to 13. Thus,
x² - 13x + 36 = 0 is equivalent to (x - 4)(x - 9) = 0, and x = 4 or x = 9.
Recall that x = y².
When x = 4: y² = 4, and so y = ±2.
When x = 9, y² = 9, and so y = ±3.
The roots are {-3, -2, 2, 3}.
The roots of the bi-quadratic equation y⁴ - 13y² + 36 = 0 are,
- 2, 2, -3, 3.
What is a polynomial?A polynomial is an algebraic expression.
A polynomial of degree n in variable x can be written as,
a₀xⁿ + a₁xⁿ⁻¹ + a₂xⁿ⁻² +...+ aₙ.
Given, y⁴ - 13y² + 36 = 0.
This a bi-quadratic polynomial.
Let, x = y².
Therefore,
x² - 13x + 36 = 0.
x² - 4x - 9x + 36 = 0.
x(x - 4) - 9(x - 4) = 0.
(x - 4)(x - 9) = 0.
x = 4 Or x = 9.
Now,
For x = 4 ⇒ y² = 4 ⇒ y = ± 2.
For x = 9 ⇒ y² = 9 ⇒ y = ± 3.
So, The roots of the biquadratic equation are - 2, 2, -3, 3.
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PLEASE HELP!
Drag the tiles to the correct boxes to complete the pairs.
Match the rational expressions to their rewritten forms.
Just Answer Please!
Answer:
1. [tex]\frac{x^2+4x-7}{x-1}[/tex]
2. [tex]\frac{x^2-2x+7}{x-1}[/tex]
3. [tex]\frac{2x^2-x-7}{x-1}[/tex]
4. [tex]\frac{2x^2-3x+7}{x-1}[/tex]
Step-by-step explanation:
1. [tex](x+5) + \frac{-2}{x-1}[/tex]
Taking LCM
[tex]=\frac{(x-1)(x+5)+(-2)}{x-1}\\ Solving:\\=frac{x(x+5)-1(x+5)-2}{x-1} \\=frac{x^2+5x-1x-5-2}{x-1} \\Adding\,\,like\,\,terms:\\=\frac{x^2+4x-7}{x-1}[/tex]
2. [tex]x-1 +\frac{6}{x-1}[/tex]
Taking LCM and solving
[tex]=\frac{(x-1)(x-1)+6}{x-1}\\=\frac{(x(x-1)-1(x-1)+6}{x-1}\\=\frac{x^2-1x-1x+1+6}{x-1}\\Adding\,\,like\,\,terms:\\=\frac{x^2-2x+7}{x-1}[/tex]
3. [tex](2x+1)+\frac{-6}{x-1}[/tex]
Taking LCM and solving:
[tex]=\frac{(2x+1)(x-1)-6}{x-1} \\=\frac{2x(x-1)+1(x-1)-6}{x-1} \\=\frac{2x^2-2x+1x-1-6}{x-1}\\Adding\,\,like\,\,terms:\\=\frac{2x^2-x-7}{x-1}[/tex]
4. [tex](2x-1)+\frac{6}{x-1}[/tex]
Taking LCM and solving:
[tex]=\frac{(2x-1)(x-1)+6}{x-1} \\=\frac{2x(x-1)-1(x-1)-6}{x-1} \\=\frac{2x^2-2x-1x+1+6}{x-1}\\Adding\,\,like\,\,terms:\\=\frac{2x^2-3x+7}{x-1}[/tex]
Answer:
I'm pretty sure that is correct.
Step-by-step explanation:
The right rectangular prism will be sliced
parallel to its base along the dashed line.
Select from the drop-down menus to correctly
describe the cross section formed by the slice.
The cross section is a Choose...
with an
area of Choose... ~
Answer:
The answer in the procedure
Step-by-step explanation:
we know that
The cross section formed by the slice is a square with the same dimensions of the base of rectangular prism
The length side of the square is 6 cm
The area of the cross section is equal to
[tex]A=b^{2}[/tex]
[tex]b=6\ cm[/tex]
substitute
[tex]A=6^{2}[/tex]
[tex]A=36\ cm^{2}[/tex]
Answer:
square and 36
Step-by-step explanation:
I took the test
Paul plans to put concrete on a rectangular portion of a driveway. The portion is 12 feet long and 6 inches high. The price of the concrete is $98.00 per cubic yard. The total cost of the concrete Paul needs is $108.89. What is the width of the driveway in feet which Paul plans to put concrete?
Answer: 5 ft
Step-by-step explanation:
Step 1: Find the volume of the driveway (in cubic yds)
[tex]\$ 108.89\div\dfrac{\$98}{yds^3}=\$ 108.89\times\dfrac{yds^3}{\$98}=\boxed{\dfrac{10}{9}yds^3}[/tex]
Step 2: Use the Volume formula (V = length × width × heighth) to find w
(convert each measurement into yds)
[tex]V=l\times w\times h\\\\\dfrac{10}{9}yds^3=12ft\bigg(\dfrac{1yd}{3ft}\bigg)\times w\times 6in\bigg(\dfrac{1ft}{12in}\bigg)\bigg(\dfrac{1yd}{3ft}\bigg)\\\\\\\dfrac{10}{9}yds^3=4yds\times w\times \dfrac{1}{6}yds\\\\\\\dfrac{10}{9}yds^3=\dfrac{2}{3}yds^2\times w\\\\\\\dfrac{3}{2yds^2}\times \dfrac{10}{9}yds^3=w\\\\\\\dfrac{5}{3}yds=w\\\\\\\dfrac{5}{3}yds\times\dfrac{3ft}{1yd}=w\\\\\\\large\boxed{5 ft=w}[/tex]
In the summer a large pool evaporates water at 15% per day. If the pool starts out with 25,700 gallons of water, which function models the pool’s loss of water?
Answer:
[tex]y=25,700(0.85)^{x}[/tex]
Step-by-step explanation:
we know that
In this problem we have a exponential function of the form
[tex]y=a(b)^{x}[/tex]
where
y ----> represent the pool’s loss of water
x ----> the number of days
a is the initial value
a=25,700 gal
b ----> is the base
r=15%=15/100=0.15
b=(1-r)=1-0.15=0.85
The function is equal to
[tex]y=25,700(0.85)^{x}[/tex]
Fracisco's game involves 3 green, 2 yellow, 4 red, and 3 black marbles. If he randomly draws three marbles from the
bag, without replacement, what is the probability that he will draw yellow, and then red, and then black?
A)1/192
B)1/72
C)3/220
D)1/55
Answer:1/55
Step-by-step explanation:
Answer: The answer is 1/55
Step-by-step explanation: because i got it right on edge. Can you mark brainliest?
40 points?With explanation
Answer:
46°
Step-by-step explanation:
Alternate angles from 46° and alternate angles are equal
Write the expression in complete factored form.
2p(n + 9) + q(n + 9) =
Answer:
(n+9) (2p+q)
Step-by-step explanation:
2p(n + 9) + q(n + 9) =
Factor out the term (n+9)
(n+9) (2p+q)
This is completely factor
How many vertical asymptotes does the graph of this function have f(x)=3/(x-11)(x+4)
Answer:
2
Step-by-step explanation:
The function is given as [tex]f(x)=\frac{3}{(x-11)(x+4)}[/tex]
Vertical asymptotes occur when the denominator is set to 0.
Thus,
(x-11)(x+4) = 0
x = 11 or x = -4
Hence, there are 2 vertical asymptotes
Answer: 2
Step-by-step explanation:
A P E X
Find the area of the circle d=8in
Answer: 64π
Step-by-step explanation: A = (d)^2 π
64 x π
64π
the lines shown below are parallel. if the green line has a slope of -1, what is the slope of the red line
Answer: -1 is the slope of the red line,
Step-by-step explanation: The slope of parallel lines are always the same. Hope this helps!
Answer:
the slope would be -1
Step-by-step explanation:
You have $20. Suppose you make x dollars in tips tomorrow at work. Which inequality must be true for you to have enough money to buy a pair of jeans after work?
Answer:
B) x +20 is greater than or less to 45
Step-by-step explanation:
Answer:
x + 20 ≥ 45
Step-by-step explanation:
You currently have : $20
You will earn : $x
Tomorrow you will end up with (20 + x) dollars
the pair of jeans cost $45, in order to afford the jeans, the amount of money that you will need must be equal or more than $45
Hence,
Money you will have tomorrow ≥ 45
or
x + 20 ≥ 45 (Answer)
The rectangular wall below is painted in 15 minutes. How many square feet per minute were painted?
The rectangular wall that is painted in 15 minute. 6.4 square feet area is painted per minute.
What is rectangle?A rectangle is a part of a quadrilateral, whose sides are parallel to each other and equal.
Given that,
The length of the rectangular wall = 12 ft.
The width of the rectangular wall = 8 ft.
The area of rectangle = 12 x 8 = 96 square feet.
Since, in 15 minutes, the rectangular wall is painted.
To find the area that painted in one minute,
Use ratio property,
15 minutes = 96 square feet painted,
1 minute = 96 / 15 square feet painted.
1 minute = 6.4 square feet.
6.4 square feet painted in 1 minute.
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Which function is shown in the graph below?
Answer:b
Step-by-step explanation: you graph each answer choice and see which one looks like the graph
Answer: The correct option is
(B) [tex]y=\left(\dfrac{1}{2}\right)^{x-3}+1.[/tex]
Step-by-step explanation: We are given to select the function that is shown in the graph.
From the graph, we know that
if the function is represented by y = f(x), then f(0) = 9. That is, the value of y at x = 0 is 9.
Option (A) : Here, the given function is
[tex]y=\left(\dfrac{1}{2}\right)^{x+3}-1.[/tex]
So, at x = 0, the value of y is given by
[tex]y(x=0)=\left(\dfrac{1}{2}\right)^{0+3}-1=\dfrac{1}{8}-1=-\dfrac{7}{8}\neq 9.[/tex]
So, this option is not correct.
Option (B) : Here, the given function is
[tex]y=\left(\dfrac{1}{2}\right)^{x-3}+1.[/tex]
So, at x = 0, the value of y is given by
[tex]y(x=0)=\left(\dfrac{1}{2}\right)^{0-3}+1=\left(\dfrac{1}{8}\right)^{-1}+1=8+1=9.[/tex]
So, this option is CORRECT.
Option (C) : Here, the given function is
[tex]y=\left(\dfrac{1}{2}\right)^{x-1}+3.[/tex]
So, at x = 0, the value of y is given by
[tex]y(x=0)=\left(\dfrac{1}{2}\right)^{0-1}+3=2+3=5\neq 9.[/tex]
So, this option is not correct.
Option (D) : Here, the given function is
[tex]y=\left(\dfrac{1}{2}\right)^{x+1}-3.[/tex]
So, at x = 0, the value of y is given by
[tex]y(x=0)=\left(\dfrac{1}{2}\right)^{0+1}-3=\dfrac{1}{2}-3=-\dfrac{5}{2}\neq 9.[/tex]
So, this option is not correct.
Thus, (B) is the correct option.
whats the difference of 2 times d minus 3
a.2
b.1 c.4
d.3
e.0
Answer:
2(d - 3) is the equation. You cannot solve for d. You can only simplify it
if f(x)=3x-1 and g(x)=x+2,find (f-g)(x)
Answer:
[tex]\large\boxed{(f-g)(x)=2x-3}[/tex]
Step-by-step explanation:
[tex](f-g)(x)=f(x)-g(x)\\\\f(x)=3x-1,\ g(x)=x+2\\\\\text{substitute:}\\\\(f-g)(x)=(3x-1)-(x+2)\\\\=3x-1-x-2\qquad\text{combine like terms}\\\\=(3x-x)+(-1-2)\\\\=2x-3[/tex]
What is the simplest form of 3square root 27a3b7
Answer:
= [tex]3ab^{\frac{2}{3} }[/tex]
Step-by-step explanation:
∛(27 a³ [tex]b^{7}[/tex])
= ∛27 · ∛a³ · ∛b³ · ∛b³ · ∛b
= 3 · a · b · b ∛b
= 3ab² [tex]b^{\frac{1}{3} }[/tex]
or
= [tex]3ab^{\frac{2}{3} }[/tex]
The function y = x^2 - 4x + 5 approximates the height, y, of a bird, and its
horizontal distance, x, as it flies from one fence post to another. All distances
are in feet. Complete the square to find and interpret the extreme value
(vertex).
Select two answers: one extreme value and one interpretation.
Height (feet)
Distance (feet)
Answer:
Option C and option D
Step-by-step explanation:
we have that
[tex]y=x^{2}-4x+5[/tex]
This is the equation of a vertical parabola open upward
The vertex is a minimum
Convert the equation into vertex form
Complete the square
[tex]y-5=x^{2}-4x[/tex]
[tex]y-5+4=x^{2}-4x+4[/tex]
[tex]y-1=x^{2}-4x+4[/tex]
[tex]y-1=(x-2)^{2}[/tex]
[tex]y=(x-2)^{2}+1[/tex] ----> equation of the parabola in vertex form
The vertex is the point (2,1)
therefore
when the bird is 2 feet away from the first fence post, it reaches its minimum height of 1 foot
Answer: C and D
Step-by-step explanation:
which of the following best describes an altitude of a three-dimensional object?
Answer:
Option C
Step-by-step explanation:
we know that
The altitude of a three-dimensional object is equal to the height of the object, is the perpendicular distance of the base to the other base or the perpendicular distance of the base to the apex of the object
therefore
A segment that is perpendicular to the planes containing the two bases
The statement which best describes an altitude of a three-dimensional object is: C. a segment that is perpendicular to the planes containing the two bases.
What is altitude?Altitude is also referred to as an elevation and it can be defined as the vertical distance (height) above the surface of a plane.
In Geometry, the altitude of a three-dimensional object is characterized by the following:
It's equal to the height of the object.It's the perpendicular distance between two bases.It's the perpendicular distance of a base to the ap-ex of an object.Read more on altitude here: https://brainly.com/question/3946367
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which expression is equivalent to...
Answer:
C
Step-by-step explanation:
Use exponents property:
[tex]\dfrac{x^a}{x^b}=x^{a-b}[/tex]
1. Note that
[tex]\dfrac{x^7}{x^{11}}=x^{7-11}=x^{-4}=\dfrac{1}{x^4}[/tex]
and
[tex]\dfrac{y^6}{y^8}=y^{6-8}=y^{-2}=\dfrac{1}{y^2}[/tex]
2. Now
[tex]\sqrt{\dfrac{55x^7y^6}{11x^{11}y^8}}=\sqrt{\dfrac{5}{x^4y^2}}=\dfrac{\sqrt{5}}{\sqrt{x^4}\sqrt{y^2}}=\dfrac{\sqrt{5}}{x^2y}[/tex]
because [tex]x>0,\ y>0[/tex]
Type the correct answer in each box. Use numerals instead of words.
You reflect triangle PQR, with coordinates P(-4, -4), Q(-1, -3), and R(-3, -1), across the x-axis, across the y-axis, and across the x-axis again to form triangle P′Q′R′.
After these reflections, the coordinates of P′ will be (,
Answer:
After these reflections, the coordinates of P′ will be (4 , -4)
Step-by-step explanation:
* Lets revise some transformation
- If point (x , y) reflected across the x-axis
∴ Its image is (x , -y)
- If point (x , y) reflected across the y-axis
∴ Its image is (-x , y)
* Lets solve the problem
- The triangle PQR with coordinates P(-4, -4), Q(-1, -3), and R(-3, -1)
- The triangle is reflected across the x-axis
∵ Δ PQR is reflected across the x-axis
∴ All y-coordinates of the vertices P, Q , R reversed their signs
∴ The new points will be (-4 , 4) , (-1 , 3) , (-3 , 1)
- The new vertices will reflected across the y-axis
∴ All x-coordinates of the new vertices reversed their signs
∴ The new points will be (4 , 4) , (1 , 3) , (3 , 1)
- The new vertices will reflected across the x-axis to form Δ P'Q'R'
∴ All y-coordinates of the new vertices reversed their signs
∴ P' = (4 , -4) , Q' = (1 , -3) , R' = (3 , -1)
* After these reflections, the coordinates of P′ will be (4 , -4)
Final answer:
After reflecting triangle PQR across the x-axis, then the y-axis, and the x-axis again, point P' will have coordinates (4, -4).
Explanation:
Reflecting a triangle across an axis involves flipping the triangle over that axis. Each reflection inverses the corresponding coordinate (x or y) of each vertex of the triangle, while the other coordinate remains the same. Starting with the first reflection across the x-axis, the y-coordinate of each point negates, but the x-coordinate remains unchanged. The second reflection is across the y-axis, which negates the x-coordinate and keeps the y-coordinate (already negated from the first reflection) the same. The third reflection across the x-axis negates the y-coordinate again, effectively returning it to its original value before the first reflection. So for point P(-4, -4), after these reflections, the new coordinates for P' will be (4, -4).
Problem
At full speed, Hal travels 600 miles in 2 hours
with the wind. The same distance against
the wind takes 3 hours.
What's the maximum speed of Hal's airplane
in still air? What's the speed of the wind?
Answer:
The maximum speed of Hal's airplane in still air is:
[tex]v= 250\ miles/h[/tex]
The speed of the wind
[tex]c = 50\ miles/h[/tex]
Step-by-step explanation:
Remember that the velocity v equals the distance d between time t.
[tex]v=\frac{d}{t}[/tex] and [tex]t*v=d[/tex]
The distance that Hal travels when traveling with the wind is:
[tex](2\ hours)(v + c) = 600[/tex] miles
Where v is the speed of Hal and c is the wind speed.
The distance when traveling against the wind is:
[tex](3\ hours)(v-c) = 600[/tex] miles
Now we solve the first equation for v
[tex](2)(v + c) = 600[/tex]
[tex]2v + 2c = 600[/tex]
[tex]2v= 600-2c[/tex]
[tex]v= 300-c[/tex]
Now we substitute the value of v in the second equation and solve for c
[tex]3((300-c)-c) = 600[/tex]
[tex]3(300-2c) = 600[/tex]
[tex]900-6c = 600[/tex]
[tex]-6c = 600-900[/tex]
[tex]-6c = -300[/tex]
[tex]6c = 300[/tex]
[tex]c = 50\ miles/h[/tex]
Then:
[tex]v= 300-(50)[/tex]
[tex]v= 250\ miles/h[/tex]
The maximum speed of Hal's airplane in still air is:
[tex]v= 250\ miles/h[/tex]
The speed of the wind
[tex]c = 50\ miles/h[/tex]
How to write (3+4i)+(8+2i) as a complex number in standard form
Answer:
Answer is 11+6i
Step-by-step explanation:
You just have to add imaginary part together and the real part. The answer will be 11+6i
Answer: 11+6i
Step-by-step explanation:
What is the slope-intercept form of the equation of the line that passes through the points (-3, 2) and (1, 5)?
A) y=3/4 x− 7/4
B) y=3/4 x- 9/2
C) y=3/4 x+ 7/2
D) y=3/4 x + 17/4
[tex]\bf (\stackrel{x_1}{-3}~,~\stackrel{y_1}{2})\qquad (\stackrel{x_2}{1}~,~\stackrel{y_2}{5}) \\\\\\ slope = m\implies \cfrac{\stackrel{rise}{ y_2- y_1}}{\stackrel{run}{ x_2- x_1}}\implies \cfrac{5-2}{1-(-3)}\implies \cfrac{3}{1+3}\implies \cfrac{3}{4}[/tex]
[tex]\bf \begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-2=\cfrac{3}{4}[x-(-3)] \implies y-2=\cfrac{3}{4}(x+3) \\\\\\ y-2=\cfrac{3}{4}x+\cfrac{9}{4}\implies y=\cfrac{3}{4}x+\cfrac{9}{4}+2\implies y=\cfrac{3}{4}x+\cfrac{17}{4}[/tex]
Which represents the solution(s) of the graphed system of equations, y = x2 + 2x – 3 and y = x – 1?
(1, 0) and (0, –1)
(–2, –3) and (1, 0)
(0, –3) and (1, 0)
(–3, –2) and (0, 1)
Answer:
Second option: (-2,-3) and (1,0)
Step-by-step explanation:
Given the system of equations [tex]\left \{ {{y = x^2 + 2x-3} \atop {y = x - 1}} \right.[/tex], you can rewrite them in this form:
[tex]x^2 + 2x-3= x - 1[/tex]
Simplify:
[tex]x^2 + 2x-3-x+1=0\\\\x^2+x-2=0[/tex]
Factor the quadratic equation. Choose two number whose sum be 1 and whose product be -2. These are: 2 and -1, then:
[tex](x+2)(x-1)=0\\\\x_1=-2\\\\x_2=1[/tex]
Substitute each value of "x" into any of the original equation to find the values of "y":
[tex]y_1= (-2) - 1=-3\\\\y_2=(1)-1=0[/tex]
Then, the solutions are:
(-2,-3) and (1,0)
ANSWER
The solutions are (-2,-3) and (1,0).
EXPLANATION
The given system has equations:
[tex]y = {x}^{2} + 2x - 3[/tex]
and
[tex]y = x - 1[/tex]
We equate both equations:
[tex] {x}^{2} + 2x - 3 = x - 1[/tex]
[tex] {x}^{2} + 2x - x - 3 + 1 = 0[/tex]
[tex] {x}^{2} + x - 2 = 0[/tex]
[tex](x - 1)(x + 2) = 0[/tex]
This implies that,
[tex]x = - 2 \: or \: x = 1[/tex]
When x=-2 , y=-2-1=-3
When x=1, y=1-1=0
The solutions are (-2,-3) and (1,0)
Which relation is a function?
Can Sb help please
A relation is a function if you associate exactly one output for every input. This means that, when you choose a value for x, there must be only one correspondent value for y. This only happens in the top-right parabola.