Exercise 52 Page 62

To solve the given equation by factoring, we will start by rewriting it, so that all terms are on the left side of the equality sign. 1/4b^2=16 ⇔ 1/4b^2-16=0 Notice that this equation follows a special pattern. It can be factored as a difference of squares. Let's factor the equation!

1/4b^2-16=0

WritePow

Write as a power

(1/2)^2b^2-4^2=0

ProdPowII

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

(1/2b)^2-4^2=0

FacDiffSquares

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

(1/2b+4)(1/2b-4)=0

Now we are ready to use the Zero Product Property.

(1/2b+4)(1/2b-4)=0

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Solve using the Zero Product Property

ZeroProdProp

Use the Zero Product Property

lc 12b+4=0 & (I) 12b-4=0 & (II)

SubEqn

(I): LHS-4=RHS-4

l 12b=- 4 12b-4=0

MultEqn

(I): LHS * 2=RHS* 2

lb=- 8 12b-4=0

AddEqn

(II): LHS+4=RHS+4

lb=- 8 12b=4

MultEqn

(II): LHS * 2=RHS* 2

lb=- 8 b=8

We found that the solutions to the given equation are b=- 8 and b=8. To check our answer, we will graph the related function, f(b)= 14b^2-16, using a calculator. Note that the calculator will use the variable x instead of b. Also, as 14=0.25, we will use the decimal instead of the fraction.

We can see that the x-intercepts are - 8 and 8. Therefore, our solutions are correct.