Example 2
Completion requirements
| Example 2 |
Convert \(y = x^2 + 8x - 3\) to vertex form by completing the square.
Step 1: Isolate the first two terms of the trinomial.
The goal is to use these two terms to build a perfect square trinomial.
\(y = \left( {x^2 + 8x} \right) - 3\)
Step 2: Build a perfect square trinomial.
Recall that the general perfect square trinomial \(a^2 + bx + c = x^2 + 2xr + r^2 \) can be factored as \(\left( {x + r} \right)^2, r = \frac{b}{2}\). Start by adding the square of one-half of \(b\), \(\left( {\left( {\frac{8}{2}} \right)^2 = 4^2 } \right)\). However, adding \(4^2\) changes the value of the function. To avoid this change, compensate by also subtracting \(4^2\).
\(y = \left( {x^2 + 8x + 4^2 - 4^2} \right) - 3\)
Step 3: Remove the subtracted value from inside the brackets.
\(\begin{array}{l}
y = \left( {x^2 + 8x + 4^2 } \right) - 3 - 4^2 \\
y = \left( {x^2 + 8x + 4^2 } \right) - 19 \end{array}\)
Step 4: Factor the perfect square trinomial.
\(y = \left( {x + 4} \right)^2 - 19\)
The result is the equation of a function expressed in vertex form.
The goal is to use these two terms to build a perfect square trinomial.
\(y = \left( {x^2 + 8x} \right) - 3\)
Step 2: Build a perfect square trinomial.
Recall that the general perfect square trinomial \(a^2 + bx + c = x^2 + 2xr + r^2 \) can be factored as \(\left( {x + r} \right)^2, r = \frac{b}{2}\). Start by adding the square of one-half of \(b\), \(\left( {\left( {\frac{8}{2}} \right)^2 = 4^2 } \right)\). However, adding \(4^2\) changes the value of the function. To avoid this change, compensate by also subtracting \(4^2\).
\(y = \left( {x^2 + 8x + 4^2 - 4^2} \right) - 3\)
Step 3: Remove the subtracted value from inside the brackets.
\(\begin{array}{l}
y = \left( {x^2 + 8x + 4^2 } \right) - 3 - 4^2 \\
y = \left( {x^2 + 8x + 4^2 } \right) - 19 \end{array}\)
Step 4: Factor the perfect square trinomial.
\(y = \left( {x + 4} \right)^2 - 19\)
The result is the equation of a function expressed in vertex form.