Practice Factoring Polynomials by Grouping

Example 6:

The product of two positive numbers is 60. Find the two numbers if one of the numbers is 4 more than the other.

Solution:

\(\begin{align*}x=\end{align*}\) one of the numbers and \(\begin{align*}x+4\end{align*}\) equals the other number. The product of these two numbers equals 60. We can write the equation.

\(\begin{align*}x(x+4)=60\end{align*}\)

Write the polynomial in standard form.

\(\begin{align*}x^2+4x&=60\\ x^2+4x-60&=0\end{align*}\)

Factor: \(\begin{align*}-60=6 \times(-10)\end{align*}\) and  \(\begin{align*}6+(-10)=-4\end{align*}\)

\(\begin{align*}-60=-6 \times 10\end{align*}\) and  \(\begin{align*}-6+10=4\end{align*}\) This is the correct choice.

The expression factors as \(\begin{align*}(x+10)(x-6)=0\end{align*}\).

Solve:

\(\begin{align*}x+10=0 && x-6& =0\\ \text{or} \\ x=-10 && x& =6\end{align*}\)

Since we are looking for positive numbers, the answer must be positive.

\(\begin{align*}x=6\end{align*}\) for one number, and  \(\begin{align*}x+4=10\end{align*}\) for the other number.

Check: \(\begin{align*}6 \cdot 10=60\end{align*}\) so the answer checks.