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The ac (cross) method

Factoring (high school)

Japanese school year: Math I

What you learn

Learn how to factor quadratic trinomials where the leading coefficient is not one by using cross-multiplication or the ac-method. This skill is helpful for factoring higher-degree polynomials and solving quadratic equations or inequalities. Prior mastery of basic factoring patterns and expansion formulas ensures smooth progress.

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Key points

Use this to factor a quadratic expression whose leading coefficient is not 1. Find factors of acac and bdbd that cross-multiply and add up to the middle coefficient ad+bcad + bc.

acx2+(ad+bc)x+bd=(ax+b)(cx+d)acx^2 + (ad + bc)x + bd = (ax + b)(cx + d)

Use this to factor quadratic expressions containing two variables, xx and yy. Focus on the numerical coefficients and apply the same cross-multiplication method.

acx2+(ad+bc)xy+bdy2=(ax+by)(cx+dy)acx^2 + (ad + bc)xy + bdy^2 = (ax + by)(cx + dy)

When the constant term is negative, its two factors must have opposite signs. Test the sign combinations systematically.

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