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Section Completing the Square
Completing the square is a powerful algebraic tool for rewriting a quadratic expression in a form that reveals key structure. While often introduced in algebra courses, the technique becomes especially important when solving differential equations using the Laplace transform. For example, when taking an inverse Laplace transform, itβs common to encounter expressions like
\(\frac{1}{(s + 2)^2 + 9}\text{.}\) Recognizing this as a completed square helps identify the corresponding time-domain function.
The idea is to transform an expression like \(x^2 + bx + c\) into a perfect square plus or minus a constant. That is,
\begin{equation*}
x^2 + bx + c = \left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 + c.
\end{equation*}
This form makes it easier to recognize shifted squares and simplify square roots or integrals.
β³οΈ Completing the Square.
Completing the square is a tool to rewrite a quadratic expression like \(x^2 + bx + c\) into the form:
\begin{gather*}
\left(x - \frac{b}{2}\right)^2 + \text{a number}
\end{gather*}
The strategy is as follows:
\begin{align*}
x^2 + bx + c
\amp = \ub{x^2 + bx + \os{(\large \text{half of } b)^2}{\boxed{\left(\frac{b}{2}\right)^2}}}_{\Large \left(x - \frac{b}{2}\right)^2}
- \ub{\os{(\large \text{half of } b)^2}{\boxed{\left(\frac{b}{2}\right)^2}} + c}_{\Large\text{a number}}
\end{align*}
Letβs walk through a few examples to get a feel for how this works, especially for later use in Laplace problems.
π Example 306 . Complete the Square of each Quadratic .
(a) \(\quad x^2 + 6x + 10\)
Solution .
Since \((\text{half of } b)^2 = (6/2)^2 = 9\text{,}\) we add and substract \(9\text{.}\)
\begin{align*}
x^2 + 6x + 10 \amp = x^2 + 6x + \boxed{9} - \boxed{9} + 10\\
\amp = (x + 3)^2 + 1
\end{align*}
(b) \(\quad x^2 - 4x + 13\)
Solution .
Since \((\text{half of } b)^2 = (4/2)^2 = 4\text{,}\) we add and substract \(4\text{.}\)
\begin{align*}
x^2 - 4x + 13 \amp = x^2 - 4x + \boxed{4} - \boxed{4} + 13\\
\amp = (x - 2)^2 + 9
\end{align*}
(c) \(\quad x^2 + 8x + 20\)
Solution .
Since \((\text{half of } b)^2 = (8/2)^2 = 16\text{,}\) we add and substract \(16\text{.}\)
\begin{align*}
x^2 + 8x + 20 \amp = x^2 + 8x + \boxed{16} - \boxed{16} + 20\\
\amp = (x + 4)^2 + 4
\end{align*}
Exercises Exercises
Exercise Group. Practice completing the square for each expression below. You can verify your answer by expanding it.
1. \(x^2 + 8x + 28\) 2. \(x^2 - 12x\) 3. \(2x^2 - 8x - 7\) 4. \(4x^2 + 8x - 65\) 5. \(-2x^2 + 20x - 47\) 6. \(-x^2 - 16x - 57\)
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