A perfect quadratic square is the product of two equal linear terms (or binomials). Previously, we considered the factorised form y=(x+n)(x+m). In a perfect square n=m and the equation is usually written as y=(x+n)². In this section we shall learn how to recongnise a perfect square when it's written in the form y=x²+bx+c. In addition, we shall study the link between perfects squares and the horizontal transformation of y=x².

The factorised form y=(x-h)² is considered for random values of h. Note, we use h rather than n or m for the square form. The minus sign has been introduced also, this can be confusing but it's there so that positive values of h corresponds to positive solutions.

The brackets (x-h) and (x-h) can be thought of as the lengths of the sides of a square and the area of the square is then y; this is shown below.

The square has been divide into 4 sub-rectangles and you can press the buttons to see which rectangles contribute to a,b, and c constants, in the equation y=ax²+bx+c.

Can you suggest a relation between the constants b and c? more specifically there are two operations that must be applied to b to transform it into c, what are they? Similarly, there are two operations to transform c into b.

Factorise the following perfect square and be mindful of the sign. Also enter the solution.

A perfect square has two identical solutions x=-h. The axis of symmetry which passes through the midpoint of the solutions is then the vertical line x=-h and vertex has the coordinate (-h,0). Use the slider to change h and observe the horizontal shift of the quadratic equation.

The orange line is the axis of symmetry. Note that, increasing h moves the quadratic equation in the negative horizontal direction, whilst decreasing h move the quadratic equation in the positive direction.

Summary

A perfect quadratic square has only one solution and touches the x-axis. The quadratic equation y=x² can be moved in the horizontal direction by change the value of h in the equation for a perfect quadratic square y=(x+h)².