Factor perfect-square trinomials

Perfect-square trinomials are quadratic expressions of the form $a^2x^2+2abx+b^2$ or $a^2x^2-2abx+b^2$ , which factor as $(ax+b)^2$ or $(ax-b)^2$ . The key reasoning is to identify the square terms, verify that the middle term is twice their corresponding factors’ product, and match the sign; this supports simplifying expressions and solving quadratic equations, without extending to higher-degree or more advanced symbolic cases.

Detailed Explanation: Factor perfect-square trinomials

A perfect-square trinomial has two square terms and a middle term that is twice the product of their square roots:

a^2x^2+2abx+b^2=(ax+b)^2

a^2x^2-2abx+b^2=(ax-b)^2.

Example

Factor:

16x^2-40x+25

Step 1: Identify the square terms.

The first term is a square:

16x^2=(4x)^2

The last term is also a square:

25=5^2

So the factors will begin as either

(4x+5)^2 \quad \text{or} \quad (4x-5)^2.

Step 2: Check the middle term.

Multiply the square roots:

(4x)(5)=20x

Double the product:

2(20x)=40x

The middle term is $(-40x)$ , so it matches the negative version.

Step 3: Write the factored form.

\boxed{16x^2-40x+25=(4x-5)^2}

You can check by expanding:

(4x-5)^2=(4x-5)(4x-5)=16x^2-40x+25.

Learn by doing: Factor perfect-square trinomials

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Polynomial Algebra X plus 1 Squared - Squared Variables with Coefficient under Square Root - Solve

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