Solving Quadratic Equations by Factoring Perfect Square Trinomials
Grade 9 students in California Reveal Math Algebra 1 learn to recognize and factor perfect square trinomials to solve quadratic equations. A perfect square trinomial matches the pattern a²+2ab+b²=(a+b)² or a²-2ab+b²=(a-b)². Students check that the first and last terms are perfect squares and the middle term equals 2ab. For example, 25x²+30x+9=0 is (5x)²+2(5x)(3)+3²=(5x+3)²=0, giving x=-3/5. If a GCF exists, like in 3x²+12x+12=0, factor it out first: 3(x+2)²=0, giving x=-2.
Key Concepts
Property If $a$ and $b$ are real numbers, some quadratic equations can be solved by factoring perfect square trinomials :.
$$a^2 + 2ab + b^2 = (a+b)^2$$ $$a^2 2ab + b^2 = (a b)^2$$.
To use this method, first write the equation in the form $=0$. Then check that the first and last terms are perfect squares ($a^2,\; b^2$) and the middle term is twice their product ($2ab$). If the trinomial matches the pattern, factor it as $(a+b)^2$ or $(a b)^2$, then solve by setting the factor equal to zero.
Common Questions
What is a perfect square trinomial?
A perfect square trinomial follows the pattern a²+2ab+b²=(a+b)² or a²-2ab+b²=(a-b)². It can be factored as the square of a binomial.
How do you check if a trinomial is a perfect square?
Verify that the first and last terms are perfect squares, then check whether the middle term equals twice the product of their square roots (2ab). If all three conditions are met, you have a perfect square trinomial.
How do you solve 25x²+30x+9=0?
Recognize it as (5x)²+2(5x)(3)+3²=(5x+3)²=0. Set 5x+3=0 and solve: x=-3/5.
How do you solve 49y²-42y+9=0?
Identify (7y)²-2(7y)(3)+3²=(7y-3)²=0. Set 7y-3=0 and solve: y=3/7.
What should you do if the trinomial has a GCF?
Factor out the GCF first. For 3x²+12x+12=0, factor out 3 to get 3(x²+4x+4)=3(x+2)²=0, then solve x+2=0 to get x=-2.
Which unit covers this factoring method?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.