Example

Subtracting y2−5yy2−4−6y−64−y2\frac{y^2-5y}{y^2-4} - \frac{6y-6}{4-y^2}

Subtract two rational expressions whose denominators are opposites, then factor and simplify:

y2−5yy2−4−6y−64−y2\frac{y^2 - 5y}{y^2 - 4} - \frac{6y - 6}{4 - y^2}

Step 1 — Recognize the opposite denominators. The denominators y2−4y^2 - 4 and 4−y24 - y^2 are opposites. Multiply the numerator and denominator of the second fraction by −1-1:

y2−5yy2−4−(6y−6)(−1)(4−y2)(−1)\frac{y^2 - 5y}{y^2 - 4} - \frac{(6y - 6)(-1)}{(4 - y^2)(-1)}

Step 2 — Simplify the second fraction. Multiplying gives −6y+6y2−4\frac{-6y + 6}{y^2 - 4}. Both fractions now share the denominator y2−4y^2 - 4:

y2−5yy2−4−−6y+6y2−4\frac{y^2 - 5y}{y^2 - 4} - \frac{-6y + 6}{y^2 - 4}

Step 3 — Subtract the numerators over the common denominator:

y2−5y−(−6y+6)y2−4\frac{y^2 - 5y - (-6y + 6)}{y^2 - 4}

Step 4 — Distribute the negative sign. Subtracting (−6y+6)(-6y + 6) reverses the signs: −(−6y)=+6y-(-6y) = +6y and −(+6)=−6-(+6) = -6:

y2−5y+6y−6y2−4\frac{y^2 - 5y + 6y - 6}{y^2 - 4}

Step 5 — Combine like terms. Group the yy-terms: −5y+6y=y-5y + 6y = y:

y2+y−6y2−4\frac{y^2 + y - 6}{y^2 - 4}

Step 6 — Factor the numerator and denominator. The numerator y2+y−6y^2 + y - 6 factors as (y+3)(y−2)(y + 3)(y - 2), since 3+(−2)=13 + (-2) = 1 and 3(−2)=−63(-2) = -6. The denominator y2−4y^2 - 4 is a difference of squares: (y−2)(y+2)(y - 2)(y + 2):

(y+3)(y−2)(y−2)(y+2)\frac{(y + 3)(y - 2)}{(y - 2)(y + 2)}

Step 7 — Simplify by removing the common factor. Cancel the shared factor (y−2)(y - 2):

y+3y+2\frac{y + 3}{y + 2}

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Updated 2026-06-18

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Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax

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