Solving Quadratic Equation by Completing the Square

If a quadratic equation of the form \({{x}^{2}}+px+q=0\) can be expressed into the form

\({{\left( x+a \right)}^{2}}=b\),

then from 1.1 Worked Example 2, it can then be solved easily by taking the square roots on both sides of the equation to obtain the solutions.

Let’s consider the expansion of \({{(x+5)}^{2}}\) using the multiplication frame.

Then \({{(x+5)}^{2}}={{x}^{2}}+10x+25\).

Note that \(10x\) is formed by adding the two terms \(5x\) together,

\({{(x+5)}^{2}}\) is a perfect square.

\({{x}^{2}}+10x\) is not a perfect square without the term, \(25\).

So, what number must be added to complete the square for \({x^2 + 10x}\)?

\(\begin{align} & {{x}^{2}}+10x=({{x}^{2}}+10x+25)-25 \\ &\qquad\qquad\, ={{(x+5)}^{2}}-25 \end{align}\)

If \({{x}^{2}}+px={{(x+a)}^{2}}-b\), then \(a=\frac{p}{2}\) and \(b={{\left( \frac{p}{2} \right)}^{2}}\) . So \({{x}^{2}}+px={{\left( x+\frac{p}{2} \right)}^{2}}-{{\left( \frac{p}{2} \right)}^{2}}\).