9-4 Solving Quadratic Equations by Graphing

9-4 Solving Quadratic Equations by 9-4 Solving Quadratic Equations by GraphingGraphing

Warm Up

1. Graph y = x2 + 4x + 3.

2. Identify the vertex and zeros of the function above.

vertex:(–2 , –1); zeros:–3, –1

Every quadratic function has a related quadratic equation. The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

y = ax2 + bx + c0 = ax2 + bx + c

When writing a quadratic function as its related quadratic equation, you replace y with 0.

One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros.

Additional Example 1A: Solving Quadratic Equations by Graphing

Solve the equation by graphing the related function.2x2 – 18 = 0 Step 1 Write the related function.

2x2 – 18 = y, or y = 2x2 + 0x – 18

Step 2 Graph the function.

• The axis of symmetry is x = 0.• The vertex is (0, –18). • Two other points (2, –10) and (3, 0)• Graph the points and reflect them across the axis of symmetry.

(3, 0) ●

x = 0

(2, –10) ●

(0, –18)●

●

●

Additional Example 1A Continued

Step 3 Find the zeros.

2x2 – 18 = 0

The zeros appear to be 3 and –3.

Substitute 3 and –3 for x in the original equation.

0 0

2(3)2 – 18 0

2(9) – 18 0 18 – 18 0

Check 2x2 – 18 = 0 2x2 – 18 = 0

The solutions of 2x2 – 18 = 0 are 3 and –3.

2(–3)2 – 18 0

2(9) – 18 0 18 – 18 0

0 0

Solve the equation by graphing the related function.

Additional Example 1B: Solving Quadratic Equations by Graphing

–12x + 18 = –2x2

Step 1 Write the related function.


y = 2x2 – 12x + 18 2x2 – 12x + 18 = 0

Use a graphing calculator.

Step 3 Find the zeros.The only zero appears to be 3. This means 3 is the only root of 2x2 – 12x + 18.


Additional Example 1C: Solving Quadratic Equations by Graphing

2x2 + 4x = –3

Step 1 Write the related function.y = 2x2 + 4x + 3


• The axis of symmetry is x = –1.• The vertex is (–1, 1). • Two other points (0, 3) and (1, 9).• Graph the points and reflect them across the axis of symmetry.

(–1, 1)

(0, 3)

(1, 9)

(–2, 3)

(–3, 9)


Additional Example 1C Continued


The function appears to have no zeros.

2x2 + 4x = –3

The equation has no real-number solutions.


Check It Out! Example 1a


x2 – 8x – 16 = 2x2


y = x2 + 8x + 16

Step 2 Graph the function.• The axis of symmetry is x = –4.• The vertex is (–4, 0). • The y-intercept is 16. • Two other points are (–3, 1) and (–2, 4).• Graph the points and reflect them across the axis of symmetry.

x = –4

(–4, 0) ●

(–3, 1) ●

(–2 , 4) ●●

●


Check It Out! Example 1a Continued


The only zero appears to be –4.

Check y = x2 + 8x + 160 (–4)2 + 8(–4) + 16 0 16 – 32 + 16 0 0

x2 – 8x – 16 = 2x2

Substitute –4 for x in the quadratic equation.


6x + 10 = –x2 Step 1 Write the related function.y = x2 + 6x + 10

Check It Out! Example 1b

Step 2 Graph the function.• The axis of symmetry is x = –3 .• The vertex is (–3 , 1). • The y-intercept is 10. • Two other points (–1, 5) and (–2, 2)• Graph the points and reflect them across the axis of symmetry.

x = –3

(–3, 1) ●

(–2, 2) ●

(–1, 5) ●

●

●


x2 + 6x + 10 = 0

Check It Out! Example 1b Continued

The equation has no real-number solutions.

Step 3 Find the zeros.The function appears to have no zeros


–x2 + 4 = 0

Check It Out! Example 1c


y = –x2 + 4

Step 2 Graph the function.Use a graphing calculator.


The function appears to have zeros at (2, 0) and (–2, 0).

Recall from Chapter 7 that a root of a polynomial is a value of the variable that makes the polynomial equal to 0. So, finding the roots of a quadratic polynomial is the same as solving the related quadratic equation.

Additional Example 2A: Finding Roots of Quadratic Polynomials

Find the roots of x2 + 4x + 3Step 1 Write the related equation.0 = x2 + 4x + 3


Step 3 Graph the related function.y = x2 + 4x + 3

• The axis of symmetry is x = –2.• The vertex is (–2, –1). • Two other points are (–3, 0) and (–4, 3)• Graph the points and reflect them across the axis of symmetry.

y = x2 + 4x + 3

(–2, –1)

(–3, 0)

(–4, 3)

Additional Example 2A Continued

Find the roots of each quadratic polynomial.


The zeros appear to be –3 and –1. This means –3 and –1 are the roots of x2 + 4x + 3.

Check 0 = x2 + 4x + 3

0 0

0 (–3)2 + 4(–3) + 3

0 9 – 12 + 3

0 = x2 + 4x + 3

0 0

0 (–1)2 + 4(–1) + 3

0 1 – 4 + 3

Additional Example 2B: Finding Roots of Quadratic Polynomials

Find the roots of x2 + x – 20Step 1 Write the related equation.0 = x2 + x – 20


Step 3 Graph the related function.y = x2 + 4x – 20

• The axis of symmetry is x = – .• The vertex is (–0.5, –20.25). • Two other points are (1, –18)

and (2, –15)• Graph the points and reflect them across the axis of symmetry.

y = x2 + 4x – 20

(–0.5, –20.25). (1, –18)

(2, –15)

Additional Example 2B Continued Find the roots of x2 + x – 20


The zeros appear to be –5 and 4. This means –5 and 4 are the roots of x2 + x – 20.

Check 0 = x2 + x – 20

0 0

0 (–5)2 – 5 – 20

0 25 – 5 – 20

0 = x2 + x – 20

0 0

0 42 + 4 – 20

0 16 + 4 – 20

Additional Example 2C: Finding Roots of Quadratic Polynomials

Find the roots of x2 – 12x + 35Step 1 Write the related equation.0 = x2 – 12x + 35 y = x2 – 12x + 35

y = x2 – 12x + 35 Step 2 Write the related function.

Step 3 Graph the related function.

• The axis of symmetry is x = 6.• The vertex is (6, –1). • Two other points (4, 3) and (5, 0)• Graph the points and reflect them across the axis of symmetry.

(6, –1).

(4, 3)

(5, 0)

Additional Example 2C Continued Find the roots of x2 – 12x + 35


The zeros appear to be 5 and 7. This means 5 and 7 are the roots of x2 – 12x + 35.

Check 0 = x2 – 12x + 35

0 0

0 52 – 12(5) + 35

0 25 – 60 + 35

0 = x2 – 12x + 35

0 0

0 72 – 12(7) + 35

0 49 – 84 + 35

Check It Out! Example 2a Find the roots of each quadratic polynomial.x2 + x – 2

Step 1 Write the related equation.0 = x2 + x – 2Step 2 Write the related function.

Step 3 Graph the related function.y = x2 + x – 2

• The axis of symmetry is x = –0.5.• The vertex is (–0.5, –2.25). • Two other points (–1, –2) and (–2, 0)• Graph the points and reflect them across the axis of symmetry.

(–0.5, –2.25).(–1, –2)(–2, 0)

y = x2 + x – 2



The zeros appear to be –2 and 1. This means –2 and 1 are the roots of x2 + x – 2.

Check 0 = x2 + x – 2

0 0

0 (–2)2 + (–2) – 2

0 4 – 2 – 2

0 = x2 + x – 2

0 0

0 12 + (1) – 2

0 1 + 1 – 2

Check It Out! Example 2a Continued

Check It Out! Example 2b Find the roots of each quadratic polynomial.9x2 – 6x + 1

Step 1 Write the related equation.0 = 9x2 – 6x + 1Step 2 Write the related function.

Step 3 Graph the related function.y = 9x2 – 6x + 1

y = 9x2 – 6x + 1

( , 0).• The axis of symmetry is x = .• The vertex is ( , 0). • Two other points (0, 1) and ( , 4)• Graph the points and reflect them across the axis of symmetry.

( , 4)

(0, 1)



Check It Out! Example 2b Continued

There appears to be one zero at . This means that is the root of 9x2 – 6x + 1.

Check 0 = 9x2 – 6x + 1

0 0

0 9( )2 – 6( ) + 1

0 1 – 2 + 1

Check It Out! Example 2c Find the roots of each quadratic polynomial.3x2 – 2x + 5

Step 1 Write the related equation.0 = 3x2 – 2x + 5

y = 3x2 – 2x + 5


Step 3 Graph the related function.y = 3x2 – 2x + 5

• The axis of symmetry is x = .• The vertex is ( , ). • Two other points (1, 6) and ( , )• Graph the points and reflect them across the axis of symmetry.

(1, 6)



Check It Out! Example 2c Continued

There appears to be no zeros. This means that there are no real roots of 3x2 – 2x + 5.

Additional Example 3: Application

A frog jumps straight up from the ground. The quadratic function f(t) = –16t2 + 12t models the frog’s height above the ground after t seconds. About how long is the frog in the air?

When the frog leaves the ground, its height is 0, and when the frog lands, its height is 0. So solve 0 = –16t2 + 12t to find the times when the frog leaves the ground and lands.

Step 1 Write the related function.0 = –16t2 + 12ty = –16t2 + 12t

Additional Example 3 Continued



Step 3 Use to estimate the zeros.The zeros appear to be 0 and 0.75.The frog leaves the ground at 0 seconds and lands at 0.75 seconds.

The frog is off the ground for about 0.75 seconds.

Check 0 = –16t2 + 12t0 –16(0.75)2 + 12(0.75)

0 –16(0.5625) + 9

0 –9 + 9

0 0

Substitute 0.75 for t in the quadratic equation.

Additional Example 3 Continued

Check It Out! Example 3

What if…? A dolphin jumps out of the water. The quadratic function y = –16x2 + 32x models the dolphin’s height above the water after x seconds. About how long is the dolphin out of the water? Check your answer.

When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is 0. So solve 0 = –16x2 + 32x to find the times when the dolphin leaves and reenters the water.

Step 1 Write the related function0 = –16x2 + 32xy = –16x2 + 32x



Step 3 Use to estimate the zeros.The zeros appear to be 0 and 2.The dolphin leaves the water at 0 seconds and reenters at 2 seconds.

The dolphin is out of the water for about 2 seconds.

Check It Out! Example 3 Continued

Check It Out! Example 3 Continued

Check 0 = –16x2 + 32x0 –16(2)2 + 32(2)

0 –16(4) + 64

0 –64 + 64

0 0

Substitute 2 for x in the quadratic equation.

Lesson Quiz Solve each equation by graphing the related function.

1. 3x2 – 12 = 0

2. x2 + 2x = 8

3. 3x – 5 = x2

4. 3x2 + 3 = 6x

5. A rocket is shot straight up from the ground. The quadratic function f(t) = –16t2 + 96t

models the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground?

2, –2

–4, 2

ø

1

6 s

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9-4 Solving Quadratic Equations by Graphing