Practice Factoring Polynomials by Grouping
Solving Real-World Problems Using Polynomial Equations
Now that we know most of the factoring strategies for quadratic polynomials, we can see how these methods apply to solving real-world problems.
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.