Exercise 64 Page 521

We want to find the roots of the given quadratic equation. In order to do that, we will factor the expression and then apply the Zero Product Property to solve the equation.

Factoring

Let's start by writing all of the terms on the left side of the equality sign. Then we will factor the expression. 2x^2+13x=24 ⇕ 2x^2+13x-24=0 Here we have a quadratic trinomial of the form ax^2+bx+c=0, where |a| ≠ 1 and there are no common factors. To factor this expression, we will rewrite the middle term, bx, as two terms. The coefficients of these two terms will be factors of ac whose sum must be b. 2x^2+13x-24=0 ⇕ 2x^2+ 13x+( - 24)=0 We have that a= 2, b= 13, and c= - 24. There are now three steps we need to follow in order to rewrite the above expression.

1. Find a c. Since we have that a= 2 and c= - 24, the value of a c is 2* ( - 24)=- 48.
2. Find factors of a c. Since a c=- 48, which is negative, we need factors of a c to have opposite signs — one positive and one negative — in order for the product to be negative. Since b= 13, which is positive, the absolute value of the positive factor will need to be greater than the absolute value of the negative factor, so that their sum is positive.

c|c|c|c 1^(st)Factor &2^(nd)Factor &Sum &Result - 1 &48 &-1 + 48 &47 - 2 &24 &-2+24 &22 - 3 & 16 & - 3 + 16 & 13

3. Rewrite bx as two terms. Now that we know which factors are the ones to be used, we can rewrite bx as two terms. 2x^2+ 13x-24=0 ⇕ 2x^2 - 3x + 16x-24=0

Finally, we will factor the last expression obtained.

Solving the Equation

Now, as we already factored the equation, we can apply the Zero Product Property to solve it.

The second root is 32. Therefore, our answer is D.