Example 1
Completion requirements
| Example 1 |
Factor the perfect square trinomial \(x^2 + 14x + 49\).
Although the examples in the previous chart are examples of perfect square trinomials where \(a = 1\), perfect square trinomials can also occur when \(a \ne 1\). However, since learning how to factor perfect square trinomials is a lead-up to a skill called completing the square, where producing perfect square trinomials with \(a = 1\) is the goal, that is where our focus lies.
Because \(a = 1\), you can apply the formula \(ax^2 + bx + c = x^2 + 2rx + r^2 = (x + r)^2, \thinspace {\rm {where}} \thinspace r = \frac{b}{2}\), recognizing that \(b = 14\).
\(\begin{align}
x^2 + 14x + 49 &=\left( {x + \frac{{14}}{2}} \right)^2 \\
&=\left( {x + 7} \right)^2 \\
\end{align}\)
\(\begin{align}
x^2 + 14x + 49 &=\left( {x + \frac{{14}}{2}} \right)^2 \\
&=\left( {x + 7} \right)^2 \\
\end{align}\)
Although the examples in the previous chart are examples of perfect square trinomials where \(a = 1\), perfect square trinomials can also occur when \(a \ne 1\). However, since learning how to factor perfect square trinomials is a lead-up to a skill called completing the square, where producing perfect square trinomials with \(a = 1\) is the goal, that is where our focus lies.