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Difficulty: MediumSolving Quadratic Equations by Factoring

Match each quadratic equation with its correct solution set by solving the equation by factoring.

  • x(x−1)=12x(x - 1) = 12x∈{−3,4}x \in \{-3, 4\}
  • 2x2+5x=32x^2 + 5x = 3x∈{−3,12}x \in \{-3, \frac{1}{2}\}
  • 3x2+8=10x3x^2 + 8 = 10xx∈{43,2}x \in \{\frac{4}{3}, 2\}
  • 2x2−24=8x2x^2 - 24 = 8xx∈{−2,6}x \in \{-2, 6\}

Answer

The equation x(x−1)=12x(x - 1) = 12 matches the solution set {−3,4}\{-3, 4\}; the equation 2x2+5x=32x^2 + 5x = 3 matches the solution set {−3,12}\{-3, \frac{1}{2}\}; the equation 3x2+8=10x3x^2 + 8 = 10x matches the solution set {43,2}\{\frac{4}{3}, 2\}; and the equation 2x2−24=8x2x^2 - 24 = 8x matches the solution set {−2,6}\{-2, 6\}.
Each quadratic equation is correctly matched to its solutions by first rewriting the equation in standard form ax2+bx+c=0ax^2 + bx + c = 0, factoring the trinomial over the integers, and then applying the zero product property to find the roots.

Step-by-Step Solution

1
Set each quadratic equation to standard form ax2+bx+c=0ax^2 + bx + c = 0 by expanding terms and moving all terms to one side.
The equations become:
1) x2−x−12=0x^2 - x - 12 = 0
2) 2x2+5x−3=02x^2 + 5x - 3 = 0
3) 3x2−10x+8=03x^2 - 10x + 8 = 0
4) 2x2−8x−24=02x^2 - 8x - 24 = 0
Before a quadratic equation can be solved by factoring, it must be set equal to zero so that the zero product property can be applied.
2
Factor each quadratic expression completely over the integers.
The factored expressions are:
1) (x−4)(x+3)=0(x - 4)(x + 3) = 0
2) (2x−1)(x+3)=0(2x - 1)(x + 3) = 0
3) (3x−4)(x−2)=0(3x - 4)(x - 2) = 0
4) 2(x−6)(x+2)=02(x - 6)(x + 2) = 0
Factoring rewrites the quadratic expressions as products of linear factors.
3
Apply the zero product property by setting each linear factor equal to zero and solving for xx.
The solution sets are:
1) x=4x = 4 or x=−3x = -3, yielding {−3,4}\{-3, 4\}
2) x=12x = \frac{1}{2} or x=−3x = -3, yielding {−3,12}\{-3, \frac{1}{2}\}
3) x=43x = \frac{4}{3} or x=2x = 2, yielding {43,2}\{\frac{4}{3}, 2\}
4) x=6x = 6 or x=−2x = -2, yielding {−2,6}\{-2, 6\}
If the product of two or more algebraic factors is zero, then at least one of the individual factors must equal zero.

Key Concept

Solving Quadratic Equations by Factoring

Alternative Method

You can verify the solution sets by substituting the values of the roots back into the original equations to check if they yield a true statement, or by using the quadratic formula x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} as an alternative algebraic method.
Estimated Time:1m 30s
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Topics

Solving Quadratic Equations by Factoring

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