Completing the square is a useful technique in algebra for transforming a quadratic expression of the form \(ax^2 + bx + c\) into a perfect square trinomial plus a constant. This technique can be especially helpful for solving quadratic equations and analyzing the properties of parabolas.
Here are the steps to complete the square for a quadratic expression:
### Step-by-Step Process:
1. **Start with the quadratic expression:**
\[ax^2 + bx + c\]
2. **If \(a \neq 1\), factor out \(a\) from the \(x^2\) and \(x\) terms:**
\[a(x^2 + \frac{b}{a}x) + c\]
3. **Transform the quadratic expression inside the parentheses into a perfect square trinomial:**
a. **Take the coefficient of the \(x\)-term inside the parentheses (which is \(\frac{b}{a}\)), and divide it by 2, then square it:**
\[
\left(\frac{\frac{b}{a}}{2}\right)^2 = \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}
\]
b. **Add and subtract this squared term inside the parentheses (to preserve the equality of the original equation):**
\[
a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right) + c
\]
4. **Combine the terms to form a perfect square trinomial and a constant:**
\[
a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c
\]
5. **Distribute \(a\) and combine constants:**
\[
a\left(x + \frac{b}{2a}\right)^2 - a\left(\frac{b^2}{4a^2}\right) + c
\]
\[
a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c
\]
6. **Combine the constants into a single constant term:**
\[
a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)
\]
By following these steps, the quadratic expression \(ax^2 + bx + c\) is transformed into the expression:
\[
a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)
\]
### Example:
Let's complete the square for the quadratic expression \(2x^2 + 8x + 5\).
1. **Original expression:**
\[2x^2 + 8x + 5\]
2. **Factor out the coefficient of \(x^2\) (which is 2):**
\[2(x^2 + 4x) + 5\]
3. **Take the coefficient of \(x\) (which is 4), divide by 2, and square it:**
\[
\left(\frac{4}{2}\right)^2 = 2^2 = 4
\]
4. **Add and subtract this squared term inside the parentheses:**
\[
2(x^2 + 4x + 4 - 4) + 5
\]
5. **Combine the terms inside the parentheses:**
\[
2((x + 2)^2 - 4) + 5
\]
6. **Distribute the 2 and combine constants:**
\[
2(x + 2)^2 - 8 + 5
\]
\[
2(x + 2)^2 - 3
\]
So, the expression \(2x^2 + 8x + 5\) is rewritten as \(2(x + 2)^2 - 3\) using the completing the square technique.
Xsquare_x_1806completing square
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