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Do these results agree with our graphs? See [link] .

This figure shows an two graphs side by side. The graph on the left side shows an upward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (-2, -1). Three points are plotted on the curve at (-3, 0), (-1, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals -2. Below the graph is the equation of the graph, y equals x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (-1, 0) and (-3, 0)”. The graph on the right side shows an downward-opening parabola graphed on the x y-coordinate plane. The x-axis of the plane runs from negative 10 to 10. The y-axis of the plane runs from negative 10 to 10. The vertex is at the point (2, 7). Three points are plotted on the curve at (-0.6, 0), (4.6, 0), and (0, 3). Also on the graph is a dashed vertical line representing the axis of symmetry. The line goes through the vertex at x equals 2. Below the graph is the equation of the graph, y equals negative x squared plus 4 x plus 3. Below that is the statement “y-intercept (0, 3)”. Below that is the statement “x-intercepts (2 plus square root of 7, 0) is approximately equal to (4.6, 0) and (2 minus square root of 7, 0) is approximately equal to (-0.6, 0).”

Find the intercepts of a parabola.

To find the intercepts of a parabola with equation y = a x 2 + b x + c :

y -intercept x -intercepts Let x = 0 and solve for y . Let y = 0 and solve for x .

Find the intercepts of the parabola y = x 2 2 x 8 .

Solution

.
To find the y -intercept, let x = 0 and solve for y . .
When x = 0 , then y = −8 .
The y -intercept is the point ( 0 , −8 ) .
.
To find the x -intercept, let y = 0 and solve for x . .
Solve by factoring. .
.

When y = 0 , then x = 4 or x = −2 . The x -intercepts are the points ( 4 , 0 ) and ( −2 , 0 ) .

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Find the intercepts of the parabola y = x 2 + 2 x 8 .

y : ( 0 , −8 ) ; x : ( −4 , 0 ) , ( 2 , 0 )

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Find the intercepts of the parabola y = x 2 4 x 12 .

y : ( 0 , −12 ) ; x : ( 6 , 0 ) , ( −2 , 0 )

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In this chapter, we have been solving quadratic equations of the form a x 2 + b x + c = 0 . We solved for x and the results were the solutions to the equation.

We are now looking at quadratic equations in two variables of the form y = a x 2 + b x + c . The graphs of these equations are parabolas. The x -intercepts of the parabolas occur where y = 0 .

For example:

Quadratic equation Quadratic equation in two variables y = x 2 2 x 15 x 2 2 x 15 = 0 ( x 5 ) ( x + 3 ) = 0 let y = 0 0 = x 2 2 x 15 0 = ( x 5 ) ( x + 3 ) x 5 = 0 x + 3 = 0 x = 5 x = −3 x 5 = 0 x + 3 = 0 x = 5 x = −3 ( 5 , 0 ) and ( −3 , 0 ) x -intercepts

The solutions of the quadratic equation are the x values of the x -intercepts.

Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the equations give the x -intercepts of the graphs, the number of x -intercepts is the same as the number of solutions.

Previously, we used the discriminant to determine the number of solutions of a quadratic equation of the form a x 2 + b x + c = 0 . Now, we can use the discriminant to tell us how many x -intercepts there are on the graph.

This figure shows three graphs side by side. The leftmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the lower right quadrant. Below the graph is the inequality b squared minus 4 a c greater than 0. Below that is the statement “Two solutions”. Below that is the statement “ Two x-intercepts”. The middle graph shows an downward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is on the x-axis. Below the graph is the equation b squared minus 4 a c equals 0. Below that is the statement “One solution”. Below that is the statement “ One x-intercept”. The rightmost graph shows an upward-opening parabola graphed on the x y-coordinate plane. The vertex of the parabola is in the upper left quadrant. Below the graph is the inequality b squared minus 4 a c less than 0. Below that is the statement “No real solutions”. Below that is the statement “ No x-intercept”.

Before you start solving the quadratic equation to find the values of the x -intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.

Find the intercepts of the parabola y = 5 x 2 + x + 4 .

Solution

.
To find the y -intercept, let x = 0 and solve for y . .
.
When x = 0 , then y = 4 .
The y -intercept is the point ( 0 , 4 ) .
.
To find the x -intercept, let y = 0 and solve for x . .
Find the value of the discriminant to predict the number of solutions and so x -intercepts. b 2 4 a c 1 2 4 5 4 1 80 −79
Since the value of the discriminant is negative, there is no real solution to the equation. There are no x -intercepts.
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Find the intercepts of the parabola y = 3 x 2 + 4 x + 4 .

y : ( 0 , 4 ) ; x : none

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Find the intercepts of the parabola y = x 2 4 x 5 .

y : ( 0 , −5 ) ; x : ( 5 , 0 ) ( −1 , 0 )

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Find the intercepts of the parabola y = 4 x 2 12 x + 9 .

Solution

.
To find the y -intercept, let x = 0 and solve for y . .
.
When x = 0 , then y = 9 .
The y -intercept is the point ( 0 , 9 ) .
.
To find the x -intercept, let y = 0 and solve for x . .
Find the value of the discriminant to predict the number of solutions and so x -intercepts. b 2 4 a c 2 2 4 4 9 144 144 0
Since the value of the discriminant is 0, there is no real solution to the equation. So there is one x -intercept.
Solve the equation by factoring the perfect square trinomial. .
Use the Zero Product Property. .
Solve for x . .
.
When y = 0 , then 3 2 = x .
The x -intercept is the point ( 3 2 , 0 ) .
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Find the intercepts of the parabola y = x 2 12 x 36 .

y : ( 0 , −36 ) ; x : ( −6 , 0 )

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Questions & Answers

Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Jose
interesting
saw
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
integer greater than 2 and less than 12
Emily Reply
2 < x < 12
Felix
I'm guessing you are doing inequalities...
Felix
Actually, translating words into algebraic expressions / equations...
Felix
hi
Darianna
hello
Mister
Eric here
Eric
He charges $125 per job. His monthly expenses are $1,600. How many jobs must he work in order to make a profit of at least $2,400?
Alicia Reply
at least 20
Ayla
what are the steps?
Alicia
6.4 jobs
Grahame
32
Grahame
1600+2400= total amount with expenses. 4000/125= number of jobs needed to make that min profit of 2400. answer is 32
Orlando
He must work 32 jobs to make a profit
POP
what is algebra
Azhar Reply
repeated addition and subtraction of the order of operations. i love algebra I'm obsessed.
Shemiah
hi
Krekar
Eric here. I'm a parent. 53 years old. I have never taken algebra. I want to learn.
Eric
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
Leanna Reply
they are 92 candies in the bag
POP
rectangular field solutions
Navin Reply
What is this?
Donna
t
muqtaar
the proudact of 3x^3-5×^2+3 and 2x^2+5x-4 in z7[x]/ is
anas Reply
?
Choli
a rock is thrown directly upward with an initial velocity of 96feet per second from a cliff 190 feet above a beach. The hight of tha rock above the beach after t second is given by the equation h=_16t^2+96t+190
Usman
Stella bought a dinette set on sale for $725. The original price was $1,299. To the nearest tenth of a percent, what was the rate of discount?
Manhwa Reply
44.19%
Scott
40.22%
Terence
44.2%
Orlando
I don't know
Donna
if you want the discounted price subtract $725 from $1299. then divide the answer by $1299. you get 0.4419... but as percent you get 44.19... but to the nearest tenth... round .19 to .2 and you get 44.2%
Orlando
you could also just divide $725/$1299 and then subtract it from 1. then you get the same answer.
Orlando
p mulripied-5 and add 30 to it
Tausif Reply
p mulripied-5 and add30
Tausif
p mulripied-5 and addto30
Tausif
how
muqtaar
Can you explain further
Monica Reply
p mulripied-5 and add to 30
Tausif
-5p+30?
Corey
p=-5+30
Jacob
How do you find divisible numbers without a calculator?
Jacob Reply
TAKE OFF THE LAST DIGIT AND MULTIPLY IT 9. SUBTRACT IT THE DIGITS YOU HAVE LEFT. IF THE ANSWER DIVIDES BY 13(OR IS ZERO), THEN YOUR ORIGINAL NUMBER WILL ALSO DIVIDE BY 13!IS DIVISIBLE BY 13
BAINAMA
When she graduates college, Linda will owe $43,000 in student loans. The interest rate on the federal loans is 4.5% and the rate on the private bank loans is 2%. The total interest she owes for one year was $1,585. What is the amount of each loan?
Ariana Reply
Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took 7 2/3 hours, what was the speed of the bus?
Kirisma Reply
66miles/hour
snigdha
How did you work it out?
Esther
s=mi/hr 2/3~0.67 s=506mi/7.67hr = ~66 mi/hr
Orlando
No; 65m/hr
albert
Practice Key Terms 6

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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