This tutorial explains the method of completing the square, a fundamental technique for rewriting quadratic expressions. This form is useful for finding the vertex of a parabola, solving quadratic equations, and calculus.
The method is based on the following algebraic identity:
\[ x^{2} + bx = \left(x + \frac{b}{2}\right)^{2} - \left(\frac{b}{2}\right)^{2} \]This formula allows us to convert a quadratic expression of the form \(x^2 + bx\) into a perfect square trinomial minus a constant.
Complete the square for \(x^{2} + 4x\).
Solution:
\[ \begin{aligned} x^{2} + 4x &= \left(x + \frac{4}{2}\right)^{2} - \left(\frac{4}{2}\right)^{2} \\ &= (x + 2)^{2} - 4 \end{aligned} \]Complete the square for \(x^{2} + 2x + 5\).
Solution:
\[ \begin{aligned} x^{2} + 2x + 5 &= \left(x + \frac{2}{2}\right)^{2} - \left(\frac{2}{2}\right)^{2} + 5 \\ &= (x + 1)^{2} - 1 + 5 \\ &= (x + 1)^{2} + 4 \end{aligned} \]Complete the square for \(2x^{2} - 12x\). Factor out the leading coefficient first.
Solution:
\[ \begin{aligned} 2x^{2} - 12x &= 2\left[ x^{2} - 6x \right] \\ &= 2\left[ \left(x + \frac{-6}{2}\right)^{2} - \left(\frac{-6}{2}\right)^{2} \right] \\ &= 2\left[ (x - 3)^{2} - 9 \right] \\ &= 2(x - 3)^{2} - 18 \end{aligned} \]Complete the square for \(-x^{2} - 10x\).
Solution:
\[ \begin{aligned} -x^{2} - 10x &= -\left[ x^{2} + 10x \right] \\ &= -\left[ \left(x + \frac{10}{2}\right)^{2} - \left(\frac{10}{2}\right)^{2} \right] \\ &= -\left[ (x + 5)^{2} - 25 \right] \\ &= -(x + 5)^{2} + 25 \end{aligned} \]Complete the square for \(-2x^{2} - 3x\).
Solution:
\[ \begin{aligned} -2x^{2} - 3x &= -2\left[ x^{2} + \frac{3}{2}x \right] \\ &= -2\left[ \left(x + \frac{3/2}{2}\right)^{2} - \left(\frac{3/2}{2}\right)^{2} \right] \\ &= -2\left[ \left(x + \frac{3}{4}\right)^{2} - \left(\frac{3}{4}\right)^{2} \right] \\ &= -2\left(x + \frac{3}{4}\right)^{2} + 2 \cdot \frac{9}{16} \\ &= -2\left(x + \frac{3}{4}\right)^{2} + \frac{9}{8} \end{aligned} \]Complete the square for \(-3x^{2} + 2x + 2\).
Solution:
\[ \begin{aligned} -3x^{2} + 2x + 2 &= -3\left[ x^{2} - \frac{2}{3}x \right] + 2 \\ &= -3\left[ \left(x + \frac{-2/3}{2}\right)^{2} - \left(\frac{-2/3}{2}\right)^{2} \right] + 2 \\ &= -3\left[ \left(x - \frac{1}{3}\right)^{2} - \left(\frac{1}{3}\right)^{2} \right] + 2 \\ &= -3\left[ \left(x - \frac{1}{3}\right)^{2} - \frac{1}{9} \right] + 2 \\ &= -3\left(x - \frac{1}{3}\right)^{2} + \frac{1}{3} + 2 \\ &= -3\left(x - \frac{1}{3}\right)^{2} + \frac{7}{3} \end{aligned} \]Complete the square for the following quadratic expressions.