Question type

Perfect square trinomials on the SAT

Q: How do you recognise a perfect square trinomial?

The first and last terms are perfect squares and the middle term is twice the product of their square roots. 16x² + 24x + 9: 16x² = (4x)², 9 = 3², and 2 × 4x × 3 = 24x.

Q: Why does the SAT give a condition like 4x + 3 < 0?

Taking a square root gives two answers, plus and minus. The condition tells you which one the question means.

A perfect square trinomial has the form $a^{2} x^{2} + 2 ab x + b^{2} = (a x + b)^{2}$ . SAT questions usually want the value of the squared expression itself, such as $4 x + 3$ , not $x$ . Rewrite the trinomial as a square, take the square root, and use any given sign condition, like $4 x + 3 < 0$ , to pick the positive or negative root.

These questions give an expression such as $16 x^{2} + 24 x + 9$ and ask you to solve for, or simplify, something built from it. The test is whether you see the square.

It appeared on one of the five official practice tests we analysed. Spotted, it takes three lines; missed, it takes a page of algebra.

On 1 of 5 official practice tests
Desmos partly helps

Why students miss it

Solving for $x$ . The question asks for $8 x + 6$ , which is $2 (4 x + 3)$ , so once you know $4 x + 3$ you are done.

Ignoring the sign condition. $(4 x + 3)^{2} = 5$ gives $4 x + 3 = \pm 5$ , and the condition $4 x + 3 < 0$ is there to pick the negative one.

How to spot it

First and last terms that are squares: $16 x^{2}$ and $9$ , or $25 p^{2}$ and $36 q^{2}$ .
A middle term equal to twice the product of their roots: $2 \cdot 4 x \cdot 3 = 24 x$ .
A condition such as " $4 x + 3 < 0$ ", and a question about a multiple of that expression.

The method

Use the square

Check that the middle term is $2 ab$ ; if so, write the trinomial as $(a + b)^{2}$ .

An equation like $(4 x + 3)^{2} = 5$: take the root: $4 x + 3 = \pm 5$ .
A sign condition is given: it picks one of the two roots.
The question asks for a multiple: scale: $8 x + 6 = 2 (4 x + 3)$ .
A fraction over the base: cancel: $\frac{( 2 x + 5 ) ^{2}}{2 x + 5} = 2 x + 5$ .

With Desmos

Desmos can solve for $x$ , but the question rarely wants $x$ .

Graph y=16x^2+24x+9 and y=5, and click the intersections to find both values of x.
Type 4x+3 with each value in place of x, keep the negative one, and double it.

This is slower than seeing the square, and two decimals invite a rounding error. Learn to recognise the pattern.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#710Adv

If $16 x^{2} + 24 x + 9 = 5$ and $4 x + 3 < 0$ , what is the value of $8 x + 6$ ?

Pick an answer, then check it.

Show the answer and explanation

The left side factors as a perfect square:

16 x^{2} + 24 x + 9 = (4 x + 3)^{2} = 5

, so

4 x + 3 = \pm 5

. Since

4 x + 3 < 0

4 x + 3 = - 5

. Then notice

8 x + 6 = 2 (4 x + 3) = - 25

2Question 2 of 2#737Adv

If $25 p^{2} - 60 pq + 36 q^{2} = 121$ and $5 p - 6 q < 0$ , what is the value of $10 p - 12 q + 7$ ?

Pick an answer, then check it.

Show the answer and explanation

Recognize the left side as a perfect square:

25 p^{2} - 60 pq + 36 q^{2} = (5 p - 6 q)^{2}

. So

(5 p - 6 q)^{2} = 121

gives

5 p - 6 q = \pm 11

, and since

5 p - 6 q < 0

, you must take

5 p - 6 q = - 11

. Then

10 p - 12 q = 2 (5 p - 6 q) = - 22

, so

10 p - 12 q + 7 = - 22 + 7 = - 15

Questions students ask

How do you recognise a perfect square trinomial?