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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

Geneva treated her parents to dinner at their favorite restaurant. The bill was $74.25. Geneva wants to leave 16 % of the total - bill as a tip. How much should the tip be?
74.25 × .16 then get the total and that will be your tip
David
$74.25 x 0.16 = $11.88 total bill: $74.25 + $11.88 = $86.13
ericka
yes and tip 16% will be $11.88
David
what is the shorter way to do it
Cesar Reply
Priam has dimes and pennies in a cup holder in his car. The total value of the coins is $4.21. The number of dimes is three less than four times the number of pennies. How many dimes and how many pennies are in the cup?
Alexandra Reply
Uno de los ángulos suplementario es 4° más que 1/3 del otro ángulo encuentra las medidas de cada uno de los angulos
Enith Reply
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Alexandra Reply
I hope this is correct, x=cooler 1 5x=cooler 2 x + 5x = 48 6x=48 ×=8 gallons 5×=40 gallons
ericka
Priam has pennies and dimes in a cup holder in his car. The total value of the coins is $4.21 . The number of dimes is three less than four times the number of pennies. How many pennies and how many dimes are in the cup?
Cecilia Reply
Arnold invested $64,000 some at 5.5% interest and the rest at 9% interest how much did he invest at each rate if he received $4500 in interest in one year
Heidi Reply
List five positive thoughts you can say to yourself that will help youapproachwordproblemswith a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
Elbert Reply
Avery and Caden have saved $27,000 towards a down payment on a house. They want to keep some of the money in a bank account that pays 2.4% annual interest and the rest in a stock fund that pays 7.2% annual interest. How much should they put into each account so that they earn 6% interest per year?
Leika Reply
324.00
Irene
1.2% of 27.000
Irene
i did 2.4%-7.2% i got 1.2%
Irene
I have 6% of 27000 = 1620 so we need to solve 2.4x +7.2y =1620
Catherine
I think Catherine is on the right track. Solve for x and y.
Scott
next bit : x=(1620-7.2y)/2.4 y=(1620-2.4x)/7.2 I think we can then put the expression on the right hand side of the "x=" into the second equation. 2.4x in the second equation can be rewritten as 2.4(rhs of first equation) I write this out tidy and get back to you...
Catherine
Darrin is hanging 200 feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length is five feet less than five times the width. Find the length and width of the fencing.
Ericka Reply
Mario invested $475 in $45 and $25 stock shares. The number of $25 shares was five less than three times the number of $45 shares. How many of each type of share did he buy?
Jawad Reply
let # of $25 shares be (x) and # of $45 shares be (y) we start with $25x + $45y=475, right? we are told the number of $25 shares is 3y-5) so plug in this for x. $25(3y-5)+$45y=$475 75y-125+45y=475 75y+45y=600 120y=600 y=5 so if #$25 shares is (3y-5) plug in y.
Joshua
will every polynomial have finite number of multiples?
cricket Reply
a=# of 10's. b=# of 20's; a+b=54; 10a + 20b=$910; a=54 -b; 10(54-b) + 20b=$910; 540-10b+20b=$910; 540+10b=$910; 10b=910-540; 10b=370; b=37; so there are 37 20's and since a+b=54, a+37=54; a=54-37=17; a=17, so 17 10's. So lets check. $740+$170=$910.
David Reply
. A cashier has 54 bills, all of which are $10 or $20 bills. The total value of the money is $910. How many of each type of bill does the cashier have?
jojo Reply
whats the coefficient of 17x
Dwayne Reply
the solution says it 14 but how i thought it would be 17 im i right or wrong is the exercise wrong
Dwayne
17
Melissa
wow the exercise told me 17x solution is 14x lmao
Dwayne
thank you
Dwayne
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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