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Question

Difficulty: EasySolving Quadratic Equations by Factoring

Match each quadratic equation with its correct set of real solutions.

  • x2−5x+6=0x^2 - 5x + 6 = 0x=2x = 2 and x=3x = 3
  • x2+5x+6=0x^2 + 5x + 6 = 0x=−3x = -3 and x=−2x = -2
  • x2−x−6=0x^2 - x - 6 = 0x=−2x = -2 and x=3x = 3

Answer

The equation x2−5x+6=0x^2 - 5x + 6 = 0 matches with the solutions x=2x = 2 and x=3x = 3. The equation x2+5x+6=0x^2 + 5x + 6 = 0 matches with the solutions x=−3x = -3 and x=−2x = -2. The equation x2−x−6=0x^2 - x - 6 = 0 matches with the solutions x=−2x = -2 and x=3x = 3.
Each equation is solved by factoring the quadratic trinomial into two binomials, then applying the zero product property to find the values of xx that make each factor zero. Specifically, x2−5x+6=0x^2 - 5x + 6 = 0 factors into (x−2)(x−3)=0(x - 2)(x - 3) = 0, yielding solutions x=2x = 2 and x=3x = 3. The equation x2+5x+6=0x^2 + 5x + 6 = 0 factors into (x+2)(x+3)=0(x + 2)(x + 3) = 0, yielding solutions x=−2x = -2 and x=−3x = -3. Finally, x2−x−6=0x^2 - x - 6 = 0 factors into (x−3)(x+2)=0(x - 3)(x + 2) = 0, yielding solutions x=3x = 3 and x=−2x = -2.

Step-by-Step Solution

1
Factor the quadratic equation x2−5x+6=0x^2 - 5x + 6 = 0.
(x−2)(x−3)=0(x - 2)(x - 3) = 0
Identify two integers that multiply to 66 and add up to −5-5. These integers are −2-2 and −3-3.
2
Solve for xx by setting each linear factor equal to zero: x−2=0x - 2 = 0 and x−3=0x - 3 = 0.
x=2x = 2 and x=3x = 3
Applying the zero product property means if the product of two numbers is zero, at least one of them must be zero.
3
Factor the quadratic equation x2+5x+6=0x^2 + 5x + 6 = 0.
(x+2)(x+3)=0(x + 2)(x + 3) = 0
Identify two integers that multiply to 66 and add up to 55. These integers are 22 and 33.
4
Solve for xx by setting each linear factor equal to zero: x+2=0x + 2 = 0 and x+3=0x + 3 = 0.
x=−2x = -2 and x=−3x = -3
Applying the zero product property gives the solutions as the negations of the terms inside the binomials.
5
Factor the quadratic equation x2−x−6=0x^2 - x - 6 = 0.
(x−3)(x+2)=0(x - 3)(x + 2) = 0
Identify two integers that multiply to −6-6 and add up to −1-1. These integers are −3-3 and 22.
6
Solve for xx by setting each linear factor equal to zero: x−3=0x - 3 = 0 and x+2=0x + 2 = 0.
x=3x = 3 and x=−2x = -2
Setting the linear factors to zero yields the roots of the equation.

Key Concept

Solving Quadratic Equations by Factoring
Estimated Time:1m 30s
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Topics

Solving Quadratic Equations by Factoring

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