10.4

10.4 Solve Quadratic Equations by Graphing

VocabularyQuadratic Equation – an equation that can be

written in the standard formwhere

Zero(s) of a function – x value(s) for which

* Zero(s) of a polynomial function and root(s) of a polynomial are the same!

Solve by Graphing (vs. Factoring)Recall Solve by Factoring:

Solve by Graphing:

Solve a quadratic equation having two solutionsExample 1

SOLUTION

STEP 1 Write the equation in standard form.

Solve by graphing.3=x2 2x–

Write original equation.3=x2 2x–

0=x2 2x– 3– Subtract 3 from each side.

STEP 2 Graph the related function . The x-intercepts are –1 and 3.y = x2 2x– 3–

Solve a quadratic equation having two solutionsExample 1

CHECK You can check –1 and 3 in the original equation.

Write original equation.3=x2 2x– 3=x2 2x–

3 3 Substitute for x.=?

=?( )21– 2– ( )1– – )32()23(

3=3 3=3 Simplify. Each solution checks.

ANSWER

The solutions of the equation are 1 and 3.3=x2 2x– –

Solve a quadratic equation having one solutionExample 2

Solve by graphing.

SOLUTION

STEP 1 Write the equation in standard form.

1=– x2 2x+

Write original equation.1=– x2 2x+

Subtract 1 from each side.0=x2 2x+ 1––

STEP 2 Graph the related function . The x-intercept is 1.

y = – x2 2x+ 1–

Solve a quadratic equation having one solutionExample 2

The solution of the equation is 1.

ANSWER

1=– x2 2x+

Solve a quadratic equation having no solutionExample 3

Solve by graphing.

SOLUTION

STEP 1 Write the equation in standard form.

4x=x2 7+

Write original equation.4x=x2 7+

0=x2 4x– 7+ Subtract 4x from each side.

STEP 2 Graph the related function The graph has no x-intercepts.

y = x2 4x– 7.+

Solve a quadratic equation having no solutionExample 3

The equation has no solution.4x=x2 7+

ANSWER

Number of Solutions of a Quadratic EquationTwo Solutions

A quadratic equation has two solutions if the graph of its related function has two x-intercepts.

One Solution

A quadratic equation has one solution if the graph of its related function has one x-intercept.

No Solution

A quadratic equation has no real solution if the graph of its related function has no x-intercepts.

The graph of the equation

what value or values of x is y 0?

ANSWER The correct answer is C.

Multiple Choice PracticeExample 4

SOLUTION

You can see from the graph that the x-intercepts are 7 and 1. So, y 0 when x 7 and x 1.= =

–= –

y = 6x+x2 7 is shown. For –=

– x = 7 only x = 1 only

x = –7 and x = 1 x = –1 and x = 7

Approximate the zeros of a quadratic function

Approximate the zeros of to the nearest tenth.

Example 5

y = 4x+x2 1+

SOLUTION

STEP 1 Graph the function There aretwo x-intercepts: one between 4

another between 1 and 0.

y = 4x+x2 1.+

–– –and 3 and

Approximate the zeros of a quadratic functionExample 5

STEP 2 Make a table of values for x-values between 4 and 3 and between 1 and 0 using anincrement of 0.1. Look for a change in the signs of the function values.

– – –

x

y

– 3.9 – 3.8 – 3.7 – 3.6 – 3.5 – 3.4 – 3.3 – 3.2 – 3.1

– 0.11 – 0.44 – 0.75 – 1.04 – 1.31 – 1.56 – 1.790.240.61

x

y

– 0.9 – 0.8 – 0.7 – 0.6 – 0.5 – 0.4 – 0.3 – 0.2 – 0.1

– 1.31 – 1.04 – 0.75 – 0.44 – 0.11 0.24 0.61– 1.56– 1.79

Approximate the zeros of a quadratic functionExample 5

In each table, the function value closest to 0 is – 0.11. So, the zeros of are about – 3.7 and about – 0.3.

y = 4x+x2 1+

ANSWER

Solve a multi-step problemExample 6

An athlete throws a shot put with an initial vertical velocity of 40 feet per second.

SPORTS

a. Write an equation that models the height h (in feet) of the shot put as a function of the time t (in seconds) after it is thrown.

b. Use the equation to find the time that the shot put is in the air.

Solve a multi-step problemExample 6

SOLUTION

a. Use the initial vertical velocity and height to write a vertical motion model.

16t2=h Vertical motion model+ vt + s–

Substitute 40 for v and 6.5 for s.16t2=h + 40t + 6.5–

b. The shot put lands when h 0. To find the time twhen h 0, solve for t.

== 16t

2=0 + 40t + 6.5–

To solve the equation, graph the related functionon a graphing calculator. Use the trace feature to find the t-intercepts.

16t2=h + 40t + 6.5–

Solve a multi-step problemExample 6

There is only one positive t-intercept. The shot put is in the air for about 2.6 seconds.

ANSWER

Relating Roots of Polynomials, Solutions of Equations, x-intercepts of Graphs, and Zeros of Functions

The Roots of the Polynomial are 2 & 6.

The Solutions of the equation are 2 & 6.

The x-intercepts of the graph of occur where y=0, so the x-intercepts are 2 and 6.

The Zeros of the function are the values of x for which y=0, so the zeros are 2 and 6.

10.4 Warm-UpSolve the equation by graphing.

1.

2.

3.