Completed Square Form Revision Notes for Leaving Cert Mathematics

Completed Square Form

The completed square form of a quadratic equation is a way to express the quadratic equation $ax^2+bx+c$ in the form :

y=a(x-h)^2+k

where $(h,k)$ is the vertex of the parabola represented by the quadratic equation.

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Completing the Square

Start with the quadratic equation in the form $ax^2+bx+c$ .
Factor out $a$ (if $a \ne 1$ ).
Add and subtract $(\tfrac{b}{2a})^2$ inside the parentheses to complete the square.
Simplify the equation to form $a(x-h)^2+k$ .

Example

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Factor out $2$ since $a \ne 1$ .

y=2(x^2+4x)+3

Add and subtract $(\tfrac{b}{2a})^2$ inside the parentheses to complete the square.

(\tfrac{b}{2a})^2=(\tfrac{4}{2(1)})^2=4

y=2(x^2+4x+4-4)+3

y=2((x+2)^2-4)+3

Simplify :

y=2(x+2)^2-5

The vertex is :

(-2,-5)