Factoring using the AC Method

How to Factor Quadratic Equations
A quadratic equation is in the form:
\(ax^2 + bx + c = 0\)
Factoring means rewriting it as a product of two binomials:
\((dx + e)(fx + g) = 0\)

The following diagram shows how to factor using a modified AC method that uses division instead of grouping. Scroll down the page for more examples and solutions of factoring using the AC method.

How to factor using a modified AC method (using division)

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Factor using the grouping method. (or AC method) Pt(1)

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Factor using the grouping method. (or AC method) Pt(2)

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AC Method (with Division)

1. Multiply \(a\) and \(c\). Let the product be \(ac\).
2. Find two numbers that multiply to \(ac\) and add up to \(b\). Let these numbers be \(m\) and \(n\).
3. Form the equation \( \frac{1}{a}(ax + m)(ax + n) = 0 \) and solve for \(x\).

AC Method (with Grouping)

1. Multiply \(a\) and \(c\). Let the product be \(ac\).
2. Find two numbers that multiply to \(ac\) and add up to \(b\). Let these numbers be \(m\) and \(n\).
3. Rewrite the middle term \(bx\) as \(mx + nx\).
4. Factor by grouping the first two terms and the last two terms.
5. Factor out the common binomial factor.

Factoring quadratics using the ac method

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Factoring a Trinomial Using the AC Method

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Tips for Factoring:

• Always look for a GCF first.
• Pay attention to the signs of the terms.
• List the factors of \(c\) (when \(a=1\)) or \(ac\) (when \(a \neq 1)\) systematically to find the correct pair.
• Check your factored form by multiplying the binomials back together to ensure you get the original quadratic.

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