Perfect Square Trinomial Calculator

Created by Anna Szczepanek, PhD
Reviewed by Wojciech Sas, PhD candidate and Jack Bowater
Last updated: Jul 13, 2021

Welcome to Omni's perfect square trinomial calculator! Here you can learn about factoring perfect square trinomials, find the perfect square trinomial formula, and go through several examples of perfect square trinomials. Do you wonder what a perfect square trinomial is? Need to learn how to factor perfect square trinomials? Wonder how to use the discriminant, Δ, to decide if your trinomial is a perfect square? Scroll down and find all the answers.

What are perfect square trinomials?

Before moving to perfect squares, let us first recall what a quadratic trinomial is in algebra, shall we? The definition is very simple: a quadratic trinomial is a polynomial of degree 2. In other words, it looks like this:

ax² + bx + c

and the real numbers a, b, c are called coefficients. We require that a ≠ 0, that is, the squared term must be present for our expression to be an actual quadratic trinomial. If a = 0 and b ≠ 0, then we have a linear binomial, and if both a and b are zero, then we end up with a simple number.

When we ask if a quadratic trinomial is a perfect square, we actually ask does a linear binomial exist such that its square gives the original trinomial.
To square a binomial means to multiply it by itself, for instance via the FOIL method.

Perfect square trinomial formula

Formally, we can write the problem of perfect square trinomials as follows:

for a trinomial

ax² + bx + c

we want to find

dx - e

such that:

(dx - e)² = ax² + bx + c.

We see that deciding whether a given trinomial is a perfect square is closely related to factoring trinomials. In fact, once you have found dx - e, you have factorized your trinomial!

Please do not confuse perfect square trinomials with [perfect square numbers](calc:3110)! Nor with [perfect pancakes](calc:1333)! 😉

How to use this perfect square trinomial calculator?

  1. Enter the coefficients a, b, c of the trinomial you have to deal with.

  2. Omni's perfect square trinomial calculator immediately finds whether your trinomial is a perfect square.

  3. If so, then it displays the linear binomial you've been looking for.

See how simple it is? Do not hesitate to use our perfect square trinomial calculator whenever you need it. "What if I ever have to do it by hand?", you ask. "How to factor perfect square trinomials by hand?" Here we are with the answer - check out the next paragraph!

How to factor perfect square trinomials?

To factor a perfect square trinomial:

  1. Check if the discriminant is zero to make sure your trinomial is a perfect square.
  2. Compute the absolute value of a and c.
  3. Evaluate the square roots √|a| and √|c|.
  4. Check the sign of a and b.
  5. Your trinomial is equal to:
    • (x√|a| + √|c|)² if a ≥ 0 and b ≥ 0;
    • -(x√|a| + √|c|)² if a < 0 and b < 0;
    • (x√|a| - √|c|)² if a ≥ 0 and b < 0 ; and
    • -(x√|a| - √|c|)² if a < 0 and b > 0.

Scroll down to find some examples of perfect square trinomials to see how this method works in practice.


How do I determine if a trinomial is a perfect square?

Compute the discriminant, that is, b² - 4 × a × c. If it is zero, then your trinomial is a perfect square.

How to make a perfect square trinomial?

You can use the short multiplication formulae:

  • (px + q)² = p²x² + 2pqx + q²
  • (px - q)² = p²x² - 2pqx + q²

What is a perfect square trinomial?

For a trinomial to be a perfect square means that we can write it as a squared linear binomial.

Is x² + 4x + 4 a perfect square trinomial?

Yes, x² + 4x + 4 is a perfect square because its discriminant is 2² - 4 × 1 × 1 = 0. To write is as a perfect square:

  1. For a = 1 and c = 4 compute √|a| = 1 and √|c| = 2.
  2. We see that a > 0 and b > 0.
  3. We use the short multiplication formula (px + q)² = p²x² + 2pqx + q² with p = 1 and q = 2.
  4. We obtain x² + 4x + 4 = (x + 2)².
  5. That's it! We've successfully used the perfect square trinomial formula! :)

Is x² + 2x + 2 a perfect square trinomial?

No, x² + 2x + 2 is not a perfect square because the discriminant is 2² - 4 × 1 × 2 = -4, which is not zero.

Anna Szczepanek, PhD
This calculator checks if the trinomial
ax² + bx + c
is a perfect square. Enter the coefficients:
Precision: 8 decimal places
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