Solving Quadratic Equations by Factoring out the GCF
Grade 9 students in California Reveal Math Algebra 1 learn to solve quadratic equations of the form ax²+bx=0 by factoring out the Greatest Common Factor. When the constant term is zero, every term contains x, so factor out x (and any shared coefficient) to rewrite as cx(dx+e)=0. Then apply the Zero Product Property: one solution will always be x=0. For example, x²-7x=0 factors to x(x-7)=0, giving x=0 or x=7; and 6x²+18x=0 factors to 6x(x+3)=0, giving x=0 or x=-3.
Key Concepts
Property For a quadratic equation where the constant term is zero, such as $$ax^2 + bx = 0$$, factor out the Greatest Common Factor (GCF) to rewrite the equation as: $$cx(dx + e) = 0$$ where $cx$ is the GCF of the terms $ax^2$ and $bx$. Then apply the Zero Product Property.
Examples Solve $x^2 7x = 0$: Factor out the GCF $x$ to get $x(x 7) = 0$. Setting each factor to zero gives $x = 0$ or $x = 7$. Solve $6x^2 + 18x = 0$: Factor out the GCF $6x$ to get $6x(x + 3) = 0$. Setting each factor to zero yields $6x = 0 \implies x = 0$ or $x + 3 = 0 \implies x = 3$.
Explanation When solving a quadratic equation that lacks a constant term, every term will contain the variable $x$. The most efficient first step is to factor out the Greatest Common Factor (GCF), which includes $x$ and any shared numerical values. Once the expression is written as a product of factors equal to zero, you can easily apply the Zero Product Property. By setting each individual factor to zero, you will find the solutions, noting that one of the roots will always be zero.
Common Questions
When do you factor out the GCF to solve a quadratic?
Factor out the GCF when the quadratic equation has no constant term — it looks like ax²+bx=0. Every term contains x, so x (and any shared numerical factor) can be factored out.
How do you solve x²-7x=0?
Factor out the GCF x: x(x-7)=0. Apply Zero Product Property: x=0 or x-7=0. Solutions are x=0 and x=7.
How do you solve 6x²+18x=0?
Factor out the GCF 6x: 6x(x+3)=0. Apply Zero Product Property: 6x=0 gives x=0; x+3=0 gives x=-3. Solutions are x=0 and x=-3.
Why is one solution always x=0 when factoring out the GCF?
Because x itself is part of the GCF. Setting x=0 satisfies the equation, giving zero as one of the roots.
What is the Zero Product Property?
The Zero Product Property states that if a product of factors equals zero, then at least one factor must equal zero. This allows you to split the factored equation into separate linear equations.
Which unit covers this factoring method in Algebra 1?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.