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Difficulty: MediumQuadratic Equations

In the equation (2x−3)(x+4)=k(2x - 3)(x + 4) = k, kk is a constant. If x=2x = 2 is a solution to the equation, what is the other solution to the equation?

  1. −92-\frac{9}{2}Answer
  2. B
    92\frac{9}{2}
  3. C
    12\frac{1}{2}
  4. D
    −9-9

Answer

−92-\frac{9}{2}
Substituting the known solution x=2x = 2 into the equation (2x−3)(x+4)=k(2x - 3)(x + 4) = k yields (2(2)−3)(2+4)=k(2(2) - 3)(2 + 4) = k, which simplifies to (1)(6)=k(1)(6) = k, so k=6k = 6. Substituting this value back into the equation gives (2x−3)(x+4)=6(2x - 3)(x + 4) = 6. Expanding the left side yields 2x2+5x−12=62x^2 + 5x - 12 = 6. Subtracting 6 from both sides places the quadratic equation in standard form: 2x2+5x−18=02x^2 + 5x - 18 = 0. Since x=2x = 2 is a root, (x−2)(x - 2) is a factor. Factoring the quadratic gives (x−2)(2x+9)=0(x - 2)(2x + 9) = 0. Setting the second factor equal to zero, 2x+9=02x + 9 = 0, yields the other solution, x=−92x = -\frac{9}{2}.

Step-by-Step Solution

1
Substitute the known solution x=2x = 2 into the equation to solve for kk.
k=6k = 6
Since x=2x = 2 is a solution, it must satisfy the equation (2(2)−3)(2+4)=k(2(2) - 3)(2 + 4) = k, which simplifies to (1)(6)=k(1)(6) = k.
2
Substitute k=6k = 6 back into the original equation, expand the binomial product, and write the quadratic equation in standard form.
2x2+5x−18=02x^2 + 5x - 18 = 0
Expanding the binomials gives 2x2+5x−12=62x^2 + 5x - 12 = 6. Subtracting 6 from both sides yields the standard form quadratic equation 2x2+5x−18=02x^2 + 5x - 18 = 0.
3
Factor the quadratic equation to find the other root.
(x−2)(2x+9)=0(x - 2)(2x + 9) = 0, yielding x=2x = 2 and x=−92x = -\frac{9}{2}
Since x=2x = 2 is a solution, (x−2)(x - 2) must be a factor. Dividing the quadratic by (x−2)(x - 2) yields the other factor, (2x+9)(2x + 9).

Key Concept

Solving quadratic equations by substituting a known root to determine constants, then rewriting and factoring the equation.

Alternative Method

Another way to find the other solution is to use the relationship between the coefficients of a quadratic equation and its roots. Once the equation is written in standard form as 2x2+5x−18=02x^2 + 5x - 18 = 0, the sum of the roots is given by −ba=−52-\frac{b}{a} = -\frac{5}{2}. Since one root is 22, the other root rr must satisfy 2+r=−522 + r = -\frac{5}{2}, which simplifies to r=−52−2=−92r = -\frac{5}{2} - 2 = -\frac{9}{2}.
Estimated Time:1m 30s
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Topics

Quadratic Equations

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