Algebra 1 Intermediate

Perfect Square Trinomials

Recognize and factor a² ± 2ab + b² = (a ± b)², or expand (a ± b)² into its trinomial form. Includes a step-by-step 3-check detection process and verification by expanding the result.

Detects PST or shows which check fails

Handles numeric and algebraic coefficients

Verification by expansion

Live

Input

Trinomial coefficients (ax² + bx + c)

x² + x +

Trinomial: x² + 6x + 9

Enter integer or decimal coefficients. The middle coefficient may be negative.

Examples

Enter a and b for (a ± b)²

( )²

Expression: (x + 3)²

Use expressions like x, 2x, 3x, or plain numbers like 5.

Examples

Result

Choose a mode, enter values, and press the button to see the factored or expanded result with step-by-step detection.

Factored Form

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First term
perfect square?

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Last term
perfect square?

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Middle = ±2ab?

Step-by-Step Solution

The Two Perfect Square Trinomial Patterns

(a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²

A perfect square trinomial is any trinomial that factors as the square of a binomial. The two patterns differ only in the sign of the middle term: positive middle → (a + b)², negative middle → (a − b)².

Examples:

x² + 6x + 9 = (x + 3)² because a = x, b = 3, and 2·x·3 = 6x ✓

4x² − 12x + 9 = (2x − 3)² because a = 2x, b = 3, and 2·2x·3 = 12x ✓ (negative)

The last term is always positive in a PST — if c is negative, it cannot be a perfect square trinomial regardless of the other terms.

Quick 3-Check Test for Perfect Square Trinomials

For a trinomial ax² + bx + c, run all three checks:

Check 1 — First term perfect square? Is a = (√a)²? The square root must be a nice integer or simple expression.
Check 2 — Last term perfect square? Is c = (√c)² with c > 0? Same requirement.
Check 3 — Middle = ±2·√a·√c? The absolute value of the middle coefficient must equal exactly 2·√a·√c.

All three checks must pass. If any fails, the trinomial is NOT a perfect square. The sign of the middle term tells you whether the factored form uses + or −.

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