Exercise 71 Page 84

To solve the given equation by factoring, we will start by identifying the values of a, b, and c. 81-y^2=0 ⇔ - 1y^2+ 0y+ 81=0 Notice that this equation follows a special pattern. It can be factored as a difference of squares. Let's factor the equation!

81-y^2=0

WritePow

Write as a power

9^2-y^2=0

FactorOutNegSignSwap

a-b=-(b-a)

-(9^2-y^2)=0

Distr

Distribute -1

y^2-9^2=0

FacDiffSquares

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

(y+9)(y-9)=0

Now we are ready to use the Zero Product Property.

(y+9)(y-9)=0

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Solve using the Zero Product Property

ZeroProdProp

Use the Zero Product Property

lcy+9=0 & (I) y-9=0 & (II)

SubEqn

(I): LHS-9=RHS-9

lcy=- 9 & (I) y-9=0 & (II)

AddEqn

(II): LHS+9=RHS+9

lcy=- 9 & (I) y=9 & (II)

We found that the solutions to the given equation are y=- 9 and y=9. To check our answer, we will graph the related function f(y)=81-y^2 using a calculator. Note that the calculator will use the variable x instead of y.

We can see that the x-intercepts are - 9 and 9. Therefore, our solutions are correct.