Solving Quadratic Equaitons

Section 3.1 beginning on page 94

The Big IdeasIn this section we will solve quadratic equations in three different ways.

• Solve By Graphing: Use the graphing calculator to find the x-intercepts (which are the solutions to the equation)

• Solve Using Square Roots: When the variable appears only once we can isolate what is being squared and find the solutions using square roots.

• Solve By Factoring: When the equation is factorable, we can use the zero product property to find the solutions to the quadratic.

Previous Knowledge:

• Properties of Square Roots• Factoring• Simplifying Radicals• Rationalizing Denominators

Core Vocabulary:

• Quadratic Equation in One Variable• Root of an Equation• Zero of a Function

Solving By GraphingExample 1:

a) b)

Enter into your graphing calculator as is.

Find the x-intercepts (zeros)

Get everything to one side and enter into your graphing calculator.

Find the x-intercepts (zeros)

Radical Review

Simplifying Radicals: If the radicand had a perfect square factor, factor it out and simplify it.

Rationalizing The Denominator: If the denominator has a radical in it, multiply the numerator and the denominator by that radical.

√72 ¿√36 √2 ¿±6 √2

√ 163 ¿ √16√3 ¿±

4

√3¿± 4 √3

3∙ √3√3

Solving Using Square RootsExample 2:

a) Step 1: Get what is being squared alone

4 𝑥2=80𝑥2=20 Step 2: Find the square root of both sides.

• Simplify the radical if possible

• Be sure to account for BOTH solutions

𝑥=√20𝑥=√ 4√5𝑥=±2√5

b)

3 𝑥2=−9𝑥2=−3

** No Real Solutions (The square of a real number cant be negative)

(𝑥± h𝑠𝑜𝑚𝑒𝑡 𝑖𝑛𝑔)2Example 2:

b) Step 1: Get what is being squared alone

Step 2: Find the square root of both sides. • Simplify the radical if possible, be sure to

account for BOTH solutions

• Rationalize the denominator (if necessary)

(𝑥+3)2=252

52∙ ∙

52

𝑥+3=√25√2

𝑥+3=±5

√2∙ √2√2

𝑥+3=± 5√22

Step 3: Get x alone.

−3−3𝑥=−3 ± 5√2

2

Zero Product PropertyThis property is why we use factoring to solve quadratic equations (when they are factorable)

0=𝑥2+6 𝑥+80=(𝑥+4)(𝑥+2)

𝑥+4=0 𝑥+2=0𝑥=−4 𝑥=−2

These are the x-intercepts, the solutions, the roots, and the zeros of the function

Solving a Quadratic Equation By Factoring

Example 3: Solve

𝑥2−4 𝑥−45=0(𝑥+5 ) (𝑥−9 )=0

𝑥+5=0 𝑥−9=0𝑥=−5 𝑥=9

Get everything to one side

Factor

Set each factor equal to zero and solve

-45 -41,-453,-155,-9

Finding the Zeros of a Quadratic Function

Example 4: Find the zeros of

24 -11-1,-24-2,-12-3,-8

Set equal to zero

Factor

Use the zero product property

0=2 𝑥2−11𝑥+12

0=2 𝑥2−3 𝑥−8 𝑥+120=𝑥 (2 𝑥−3 )−4 (2 𝑥−3)0=(2 𝑥−3 )(𝑥−4)

2 𝑥−3=0 𝑥−4=0

𝑥=32 𝑥=4

2 𝑥=3

Solving a Multi-Step Problem

Annual Revenue

Number of $1 increases in price

𝑟 (𝑥 )=¿ (48,000−2000 𝑥)(20+1𝑥)

Continued…

𝑥=24𝑟 (𝑥 )=(48000−2000𝑥 )(20+𝑥 ) To find the maximum, find the y-value of

the vertex (section 2.2)𝑥=−20

𝑥=24+(−20)

2 𝑥=2

𝑟 (2 )=(48000−2000 (2))(20+(2))𝑟 (2 )=968,000

To maximize revenue each subscription should cost $22 (20 + x) and the maximum revenue would be $968,000

PracticeSolve the equation by graphing.

1) 2) 3)

Solve the equation using square roots.

4) 5) 6)

Solve the equation by factoring.

7) 8)

Find the zero(s) of the function.

9) 10)

1) 2) 3) No real solutions4) 5) 6) No real solutions7) 8) 9) 10)