Completing the Square

Completing the square is useful method for solving a number of problems and finding new information about a quadratic expression. In general, completing the square changes an expression with two terms involving $x$ into one where $x$ appears only once.

$$x^2+bx+c={(x+p)}^2+q$$

$p$ and $q$ can be used to locate the vertex of the parabola, i.e., $(\mathit{-p},q)$. Alternatively the completed expression can be used in an equation to find the solutions.

In order to convert the lefthand side to the right we need only concentrate on the first two terms, $x^2+bx$, to start with.

1. Write down $(x+$;
2. then $b$ divided by two;
3. followed by ${)}^2$;
4. now subtract ${\left(\frac{b}{2}\right)}^2$.

Having complete this method we then have to substitute back into the original problem
and, perhaps, simplify. There is a proof of this method.

Example 1:

$$x^2+4x$$ $$={(x+\frac{4}{2})}^2-{\left(\frac{4}{2}\right)}^2$$ $$={(x+2)}^2-2$$

Example 2:

$$x^2-6x+10$$ $$={(x-\frac{6}{2})}^2-{\left(\frac{6}{2}\right)}^2+10$$ $$={(x-3)}^2-9+10$$ $$={(x-3)}^2+1$$

Example 3:

$$2x^2-9x-1$$ $$=2(x^2-\frac{9}{2}x)-1$$ $$=2({(x-\frac{9}{4})}^2-{\left(\frac{9}{4}\right)}^2)-1$$ $$=2{(x-\frac{9}{4})}^2-2\left(\frac{81}{16}\right)-1$$ $$=2{(x-\frac{9}{4})}^2-\frac{81}{8}-1$$ $$=2{(x-\frac{9}{4})}^2-\frac{89}{8}$$

Completing the Square (CtS) Video

Practice Questions

Textbook Exercises

Textbook Exercises - solving with CtS

Quadratic graph (CtS) Video

(from corbettmaths.com)