Chapter 9, Problem 32SGR

To determine

To solve the given equation by factoring.

Answer to Problem 32SGR

The value of g is -15 and -1 .

Explanation of Solution

Given information :

The equation provided is ${\left(g+8\right)}^2=49$ .

Formula used :

The given equation can be solved by factoring. To do so, break up the midpoint to arrive at two factors. Then use the zero-product property to equate each factor in order to get the roots.

Calculation :

  ${\left(g+8\right)}^2=49$

Given equation.

  $=>g^2+16g+64-49=0$

On expanding and subtracting 49 from both sides.

  $=>g^2+16g+15=0$

Equation to be solved.

  $=>g^2+15g+g+15=0$

Breaking the middle term that when multiplied, provides that values of $a{x}^2\times c$ .

  $=>g\left(g+15\right)+1\left(g+15\right)=0$

Arranging the common factors.

  $=>\left(g+1\right)\left(g+15\right)=0$

The two factors of the function.

Now, the first root is:

  $g+1=0$

Using zero-product property.

  $g=-1$

The second root is:

  $g+15=0$

Using zero-product property.

  $g=-15$

Thus, the values of g are -15 and -1 .

To check the solutions, put them in the given equation and verify that left-hand and right-hand sides are equal.

For $g=-1$

Putting the value in the equation.

  ${\left(-1\right)}^2+16\left(-1\right)+15=0$

Simplifying the expression.

  $=>1-16+15=0$

Hence this result is a success.

Now, for $g=-15$

  ${\left(-15\right)}^2+16\left(-15\right)+15=0$

Putting the value in the equation.

  $=>225-240+15=0$

Simplifying the expression.

  $=>240-240=0$

Hence this result is a success.