Question

What are the solutions to (x+7)2=81?

A. –74 and 88
B. –2 and 16
C. –88 and 74
D. –16 and 2

Answer 1

Step-by-step explanation:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

(x-7)^2-(81)=0

Step by step solution :

STEP

1

:

1.1 Evaluate : (x-7)2 = x2-14x+49

Trying to factor by splitting the middle term

1.2 Factoring x2-14x-32

The first term is, x2 its coefficient is 1 .

The middle term is, -14x its coefficient is -14 .

The last term, "the constant", is -32

Step-1 : Multiply the coefficient of the first term by the constant 1 • -32 = -32

Step-2 : Find two factors of -32 whose sum equals the coefficient of the middle term, which is -14 .

-32 + 1 = -31

-16 + 2 = -14 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -16 and 2

x2 - 16x + 2x - 32

Step-4 : Add up the first 2 terms, pulling out like factors :

x • (x-16)

Add up the last 2 terms, pulling out common factors :

2 • (x-16)

Step-5 : Add up the four terms of step 4 :

(x+2) • (x-16)

Which is the desired factorization

Equation at the end of step

1

:

(x + 2) • (x - 16) = 0

STEP

2

:

Theory - Roots of a product

2.1 A product of several terms equals zero.

When a product of two or more terms equals zero, then at least one of the terms must be zero.

We shall now solve each term = 0 separately

In other words, we are going to solve as many equations as there are terms in the product

Any solution of term = 0 solves product = 0 as well.

Solving a Single Variable Equation:

2.2 Solve : x+2 = 0

Subtract 2 from both sides of the equation :

x = -2

Solving a Single Variable Equation:

2.3 Solve : x-16 = 0

Add 16 to both sides of the equation :

x = 16