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McGraw Hill Integrated II, 2012

MH

McGraw Hill Integrated II, 2012 View details

2. Solving Quadratic Equations by Graphing

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Exercise 59 Page 110

Recall the formula for factoring a perfect square trinomial.

- 3/2, 5/2

Practice makes perfect

We will solve the equation and then check the solutions.

Solving the Equation

To solve the given quadratic equation, we will start by paying close attention to the terms on the left-hand side. We can see that the first and last terms are perfect squares, and that the middle term is twice the product of their square roots. This means that, the left-hand side is a perfect square trinomial and we can factor it following the corresponding formula. a^2-2ab+b^2=(a-b)^2 Let's solve the equation by factoring the perfect square trinomial on the left-hand side.

4x^2-4x+1=16

Write as a power

2^2x^2-4x+1^2=16

a^m b^m = (a b)^m

(2x)^2-4x+1^2=16

SplitIntoFactors

Split into factors

(2x)^2-2(2)x+1^2=16

AssociativePropMult

Associative Property of Multiplication

(2x)^2-2(2x)+1^2=16

Identity Property of Multiplication

(2x)^2-2(2x)(1)+1^2=16

FacNegPerfectSquare

a^2-2ab+b^2=(a-b)^2

(2x-1)^2=16

sqrt(LHS)=sqrt(RHS)

sqrt((2x-1)^2)=sqrt(16)

sqrt(a^2)=± a

2x-1=± 4

LHS+1=RHS+1

2x=1± 4

.LHS /2.=.RHS /2.

x=1± 4/2

We will now find the first and second solutions by using the positive and negative signs.

x=1± 4/2
x=1+4/2	x=1-4/2
x=5/2	x=- 3/2

We found that the solutions to the equation are - 32 and 52.

Checking the Solutions

We can check the solutions by substituting them for x in the given equation. Let's start with 52.

4x^2-4x+1=16

x= 2.5

4( 5/2)^2-4( 5/2)+1? =16

(a/b)^m=a^m/b^m

4(25/4)-4(5/2)+1? =16

MoveLeftFacToNum

a*b/c= a* b/c

100/4-20/2+1? =16

Calculate quotient

25-10+1

Add and subtract terms

16=16 ✓

Since substituting and simplifying created a true statement, we know that 52 is a solution to the equation. Let's check if - 32 is also a solution.

4x^2-4x+1=16

x= - 3/2

4( - 3/2)^2-4( - 3/2)+1? =16

NegBaseToPosPow

(- a)^2 = a^2

4(3/2)^2-4(- 3/2)+1? =16

(a/b)^m=a^m/b^m

4(9/4)-4(- 3/2)+1? =16

a(- b)=- a * b

4(9/4)-(- 4)(3/2)+1? =16

MoveLeftFacToNum

a*b/c= a* b/c

36/4-(- 12/2)+1? =16

MoveNegNumToFrac

Put minus sign in front of fraction

36/4-(- 12/2)+1? =16

a-(- b)=a+b

36/4+12/2+1? =16

a/b=a * 2/b * 2

36/4+24/4+1? =16

Rewrite 1 as 4/4

36/4+24/4+4/4? =16

Add fractions

64/4? =16

Calculate quotient

16=16 ✓

We created another true statement, so we know that - 32 is also a solution to the equation.

Subchapter links

Exercises p.108-110