To factor a perfect square trinomial, follow these steps:
- Identify a perfect square trinomial:
The trinomial should be in the form a2x2+2abx+b2a^2x^2+2abx+b^2a2x2+2abx+b2 or a2x2−2abx+b2a^2x^2-2abx+b^2a2x2−2abx+b2, where the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms
- Find the square roots of the first and last terms:
- The square root of the first term a2x2a^2x^2a2x2 is axaxax.
- The square root of the last term b2b^2b2 is bbb
- Check the middle term:
Confirm that the middle term is ±2abx\pm 2abx±2abx. The sign of the middle term determines whether the binomial factor uses addition or subtraction
- Write the factored form:
- If the middle term is positive, factor as (ax+b)2(ax+b)^2(ax+b)2.
- If the middle term is negative, factor as (ax−b)2(ax-b)^2(ax−b)2
Example
Factor 9x2+30x+259x^2+30x+259x2+30x+25:
- 9x2=(3x)29x^2=(3x)^29x2=(3x)2
- 25=5225=5^225=52
- The middle term 30x=2×3x×530x=2\times 3x\times 530x=2×3x×5
- Since the middle term is positive, the factorization is (3x+5)2(3x+5)^2(3x+5)2
Summary
- Confirm first and last terms are perfect squares.
- Confirm middle term equals twice the product of the square roots of first and last terms.
- Use the sign of the middle term to decide the sign in the binomial.
- Write the factorization as the square of a binomial.
This method reverses the process of squaring a binomial and is a quick way to factor perfect square trinomials
