Perfect Squares

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

A perfect-square trinomial has square outer terms and a middle term twice the product of their square roots. That middle-term check distinguishes an identity from a tempting but false pattern.

Method

A^2+2AB+B^2=(A+B)^2.

A^2-2AB+B^2=(A-B)^2.

A repeated linear factor gives a repeated zero. Its graph touches the horizontal axis rather than crossing there.

Worked example

In $4x^2-12x+9$ , the outer terms are $(2x)^2$ and $3^2$ , and $-12x=-2(2x)(3)$ . Therefore

4x^2-12x+9=(2x-3)^2.

Its only zero is $x=3/2$ , with multiplicity two; its value cannot be negative.

Common mistake

$A^2+B^2$ is missing the mixed term and is not $(A+B)^2$ .

Check your understanding

Factor $9y^2+24y+16$ .

Show answer

$(3y+4)^2$ . The middle term is $2(3y)(4)$ .

Explore

Factors and zeros

Try this. Move the roots together and apart. Match the expanded coefficients to their sum and product, then compare touching with crossing the axis.

First root-2Second root2

(x − (-2))(x − (2)) = x² − (0)x + (-4). Zeros: -2, 2. Simple roots: the graph crosses at each zero.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which expression is a perfect square?

$x^2+6x+9$

$x^2+9$

$x^2+3x+9$

Hint 1 · Find a starting point

A square (x+a)² includes a middle term 2ax.

Hint 2 · Take the next step

If a² = 9 and a = 3, then 2ax = 6x.

Show the reasoning

Answer: $x^2+6x+9$

x²+6x+9 = (x+3)².

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

Optional marks, not a grade. Saved in this browser only. Open notebook →