Completing The Square

Completing the square is a process of which is able to give a quadratic an alternate form, the completed square form can be used to go about solving an equation, or used to find the turning point of a graph.

Form

The completed square form is: \((x + a)^2 + b\).

Convert a Quadratic

In having a quadratic, in the form of \(x^2 + bx + c\), you can use the following to convert it to a completed square form.

\[ x^2 + bx + c = (x + \frac{b}{2})^2 - \frac{b}{2}^2 + c \]

Solve a Quadratic

Having completed the square for a quadratic, you can go about solving it rather simply.

\[ \begin{align} x^2 + bx + c & = d \\ (x + \frac{b}{2})^2 - \frac{b}{2}^2 + c & = d \end{align} \]

First, you will want to go about changing the form of the equation to equal \((x + a)^2\). Having done this, you can go about square rooting the equation, to have it equal \(x + a\).

\[ \begin{align} (x + \frac{b}{2})^2 & = \frac{b}{2}^2 - c + d \\ x + \frac{b}{2} & = \pm\sqrt{\frac{b}{2}^2 - c + d} \end{align} \]

The equation can now go about being made equal to \(x\), bearing in mind there is going to be two solutions.

\[ x = \pm\sqrt{\frac{b}{2}^2 - c + d} - \frac{b}{2} \]

Example

To supplement the above given method, here is an example of solving a quadratic using completing the square.

\[ \begin{align} x^2 + 6x + 5 & = 2 \\ (x + \frac{6}{2})^2 - \frac{6}{2}^2 + 5 & = 2 \\ (x + 3)^2 - 9 + 5 & = 2 \\ (x + 3)^2 - 4 & = 2 \\ (x + 3)^2 & = 6 \\ x + 3 & = \pm\sqrt{6} \\ x & = \pm\sqrt{6} - 3 \end{align} \]

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