# algebra 1b chapter 9 solving quadratic equations by graphing

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Algebra 1BAlgebra 1BChapter 9Chapter 9

Solving Quadratic EquationsBy Graphing

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.

Step 2 Graph the 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.

Solve the equation by graphing the related function.

Additional Example 1C: Solving Quadratic Equations by Graphing

2x2 + 4x = –3

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

Step 2 Graph the function.

• 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)

Solve the equation by graphing the related function.

Additional Example 1C Continued

Step 3 Find the zeros.

The function appears to have no zeros.

2x2 + 4x = –3

The equation has no real-number solutions.

Solve the equation by graphing the related function.

In Your Notes! Example 1a

Solve the equation by graphing the related function.

x2 – 8x – 16 = 2x2

Step 1 Write the related function.

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) ●●

●

Solve the equation by graphing the related function.

In Your Notes! Example 1a Continued

Step 3 Find the zeros.

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.

Solve the equation by graphing the related function.

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

In Your Notes! 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) ●

●

●

Solve the equation by graphing the related function.

x2 + 6x + 10 = 0

In Your Notes! Example 1b Continued

The equation has no real-number solutions.

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

Solve the equation by graphing the related function.

–x2 + 4 = 0

In Your Notes! Example 1c

Step 1 Write the related function.

y = –x2 + 4

Step 2 Graph the function.Use a graphing calculator.

Step 3 Find the zeros.

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

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 2 Write the related function.

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.

Step 4 Find the zeros.

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 2 Write the related function.

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

Step 4 Find the zeros.

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

Step 4 Find the zeros.

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

In Your Notes! 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

Find the roots of each quadratic polynomial.

Step 4 Find the zeros.

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

In Your Notes! Example 2a Continued

In Your Notes! 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)

Find the roots of each quadratic polynomial.

Step 4 Find the zeros.

In Your Notes! 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

In Your Notes! 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 2 Write the related function.

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)

Find the roots of each quadratic polynomial.

Step 4 Find the zeros.

In Your Notes! Example 2c Continued

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