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Identify two numbers that multiply to the constant term (-6) and add up to the coefficient of the x term (-1).

Mathematics

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Answer

x=3,x=−2x = 3, x = -2

Step-by-step solution

5 steps

To solve the quadratic equation x2−x−6=0x^2 - x - 6 = 0 by factoring:

  1. Identify two numbers that multiply to the constant term (−6-6) and add up to the coefficient of the xx term (−1-1).

    The two numbers are 2 and −3-3, because 2×(−3)=−62 \times (-3) = -6 and 2+(−3)=−12 + (-3) = -1.

  2. Rewrite the middle term using these two numbers.

    x2+2x−3x−6=0x^2 + 2x - 3x - 6 = 0

  3. Factor by grouping.

    Group the first two terms and the last two terms.

    (x2+2x)−(3x+6)=0(x^2 + 2x) - (3x + 6) = 0

    Factor out the common term from each group.

    x(x+2)−3(x+2)=0x(x + 2) - 3(x + 2) = 0

  4. Factor out the common binomial (x+2)(x+2).

    (x−3)(x+2)=0(x - 3)(x + 2) = 0

  5. Set each factor equal to zero and solve for xx.

    x−3=0orx+2=0x - 3 = 0 \quad \text{or} \quad x + 2 = 0 x=3orx=−2x = 3 \quad \text{or} \quad x = -2

    The solutions are x=3x = 3 and x=−2x = -2.

    The final answers are x=3,x=−2x = 3, x = -2.

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