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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