Chapter 4.3, Problem 79HP

To determine

To describe: The correct roots of the polynomial $-12x^2+5x+2=0$ calculated by Gwen and Morgan.

Answer to Problem 79HP

Morgan calculated the roots correctly.

Explanation of Solution

Given information:

The roots of polynomial $-12x^2+5x+2=0$ by Gwen and Morgan.

Consider the factorization of polynomial $-12x^2+5x+2=0$ by Gwen and Morgan.

When Gwen did the calculations in the third step, he took a negative sign common so, sign will change in the parenthesis.

On the other hand Morgan did the correct calculations. He split the middle term and then applied the zero property.

Recall that to factor a polynomial split the middle term in such a way that sum of two numbers is the middle term and product of two numbers is same is product of first and last term.

For the above polynomial $8$ and $-3$ are two numbers such that their sum is $5$ and product is $-24$ .

Apply it, first the group the similar terms,

  $\begin{array}{c}-12x^2+5x+2=0\\ -12x^2+8x-3x+2=0\\ 4x\left(-3x+2\right)+\left(-3x+2\right)=0\\ \left(4x+1\right)\left(-3x+2\right)=0\end{array}$

Apply the zero property,

Either $\left(4x+1\right)=0\text{ or }\left(-3x+2\right)=0$ .

Therefore, $x=-\frac{1}{4}\text{ or }\frac{2}{3}$ .