A quadratic curve is usually written in the form $$ \begin{equation}\begin{aligned} y=ax^2+bx+c\\ \end{aligned}\end{equation} $$
This can be rewritten in the form $a(x+h)^2+k$ in order to extract important information such as:
- The axis of symmetry $x=-h$
- The maximum/minimum value $y=k$ and
- The turning point $(-h,k)$.
Example
Complete the square for the curve $2x^2+5x+6$.
Solution First, we pair off the terms in $x$: $$ \begin{equation}\begin{aligned} (2x^2+5x)+6\\ \end{aligned}\end{equation} $$
Next we factorize out the coefficient of $x^2$: $$ \begin{equation}\begin{aligned} 2(x^2+\frac{5}{2}x)+6\\ \end{aligned}\end{equation} $$
Now we add the square of half of the coefficient of $x$ to the bracket, subtracting the equivalent from the outside of the bracket to balance the change: $$ \begin{equation}\begin{aligned} \textcolor{forestgreen}{2}(x^2+\frac{5}{2}x+\textcolor{crimson}{[\frac{5}{4}]^2})+6\textcolor{forestgreen}{-2}\textcolor{crimson}{[\frac{5}{4}]^2}\\ \end{aligned}\end{equation} $$
Notice that by adding the $[\frac{5}{4}]^2$ to the bracket, we effectively added $2[\frac{5}{4}]^2$ overall to the expression. Hence, we had to subtract $2[\frac{5}{4}]^2$ in order to balance this increase.
We now apply the perfect square factorization to the bracket: $$ \begin{equation}\begin{aligned} 2(x+\frac{5}{4})^2+6-2[\frac{5}{4}]^2\\ \end{aligned}\end{equation} $$
And simplify the terms after the bracket: $$ \begin{equation}\begin{aligned} &2(x+\frac{5}{4})^2+6-2[\frac{25}{16}]\\ &2(x+\frac{5}{4})^2+6-\frac{25}{8}\\ &2(x+\frac{5}{4})^2+\frac{48-25}{8}\\ &2(x+\frac{5}{4})^2-\frac{23}{8}\\ \end{aligned}\end{equation} $$
We thus have the quadratic in the form $a(x+h)^2+k$ where $h=\frac{5}{4}$ and $k=\frac{23}{8}$.
Created using natural intelligence