Solving Quadratic Equations by Factoring: ac Method
Grade 9 students in California Reveal Math Algebra 1 learn the ac method for factoring and solving quadratic equations ax²+bx+c=0 where the leading coefficient a≠1. The method requires finding two numbers m and n such that m·n=ac and m+n=b, then splitting the middle term and factoring by grouping. For example, solving 2x²+5x+3=0: ac=6, find m=2 and n=3, rewrite as 2x²+2x+3x+3=0, factor to (2x+3)(x+1)=0, giving x=-3/2 or x=-1. The Zero Product Property then extracts each solution from the factored form.
Key Concepts
To solve a quadratic equation $ax^2 + bx + c = 0$ where $a \neq 1$ using the ac method :.
1. Set the equation equal to zero. 2. Find the product $ac$. 3. Find two numbers $m$ and $n$ such that $m \cdot n = ac$ and $m + n = b$. 4. Rewrite the middle term: $ax^2 + mx + nx + c = 0$. 5. Factor by grouping, then apply the Zero Product Property to solve.
Common Questions
What is the ac method for factoring quadratic equations?
The ac method factors ax²+bx+c=0 by finding two numbers m and n where m·n=ac and m+n=b. You then rewrite the middle term using m and n, factor by grouping, and apply the Zero Product Property to solve.
How do you solve 2x²+5x+3=0 using the ac method?
Compute ac=2·3=6. Find m and n where m·n=6 and m+n=5: m=2, n=3. Rewrite: 2x²+2x+3x+3=0. Factor: 2x(x+1)+3(x+1)=0 → (2x+3)(x+1)=0. Solutions: x=-3/2 or x=-1.
What is the Zero Product Property?
The Zero Product Property states that if a product of two factors equals zero, then at least one of the factors must equal zero. This allows you to set each factor equal to zero and solve separately.
How do you solve 3x²-7x+2=0 using the ac method?
Compute ac=6. Find m=-6 and n=-1 since (-6)(-1)=6 and -6+(-1)=-7. Rewrite: 3x²-6x-x+2=0. Factor: 3x(x-2)-1(x-2)=0 → (3x-1)(x-2)=0. Solutions: x=1/3 or x=2.
When should you use the ac method instead of simple factoring?
Use the ac method when the leading coefficient a is not equal to 1, making simple inspection difficult. It provides a systematic way to factor quadratics that cannot be easily factored by trial and error.
Which unit covers the ac method in Algebra 1?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.