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Difficulty: MediumSolving Quadratic Equations by Factoring

A certain value xx satisfies the relationship where the square of the difference between xx and 44 is equal to 1616 decreased by 33 times xx. If x1x_1 and x2x_2 are the two real solutions to this relationship, with x1>x2x_1 > x_2, what is the value of 3x1−2x23x_1 - 2x_2?

  1. A
    3
  2. B
    6
  3. C
    10
  4. 15Answer
  5. E
    33

Answer

The value of the expression is 15.
The value 15 is correct. First, translate the relationship into the algebraic equation (x−4)2=16−3x(x-4)^2 = 16 - 3x. Expanding the left side yields x2−8x+16=16−3xx^2 - 8x + 16 = 16 - 3x. To solve the quadratic equation by factoring, rearrange the terms to set one side to zero: subtract 16 and add 3x3x to both sides, which simplifies to x2−5x=0x^2 - 5x = 0. Factoring the left-hand side gives x(x−5)=0x(x - 5) = 0. The solutions are x=5x = 5 and x=0x = 0. Since the problem defines x1>x2x_1 > x_2, the larger solution is x1=5x_1 = 5 and the smaller solution is x2=0x_2 = 0. Substituting these values into the expression 3x1−2x23x_1 - 2x_2 yields 3(5)−2(0)=153(5) - 2(0) = 15.

Step-by-Step Solution

1
Translate the verbal description into a mathematical equation.
(x−4)2=16−3x(x - 4)^2 = 16 - 3x
The 'square of the difference between xx and 44' is represented as (x−4)2(x-4)^2, and '16 decreased by 3 times xx' is represented as 16−3x16 - 3x.
2
Expand the squared binomial on the left side of the equation.
x2−8x+16=16−3xx^2 - 8x + 16 = 16 - 3x
Using the binomial expansion formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2, expand (x−4)2(x-4)^2 to x2−2(4)(x)+42=x2−8x+16x^2 - 2(4)(x) + 4^2 = x^2 - 8x + 16.
3
Rearrange the equation to set one side to zero.
x2−5x=0x^2 - 5x = 0
Add 3x3x and subtract 1616 from both sides of the equation to gather all terms on one side: x2−8x+3x+16−16=0x^2 - 8x + 3x + 16 - 16 = 0, which simplifies to x2−5x=0x^2 - 5x = 0.
4
Factor the quadratic equation.
x(x−5)=0x(x - 5) = 0
Factor out the greatest common factor, which is xx.
5
Solve for the roots of the equation.
x=0x = 0 or x=5x = 5
Set each factor to zero: x=0x = 0 or x−5=0x - 5 = 0.
6
Identify the values of x1x_1 and x2x_2 and calculate the required expression.
x1=5x_1 = 5, x2=0x_2 = 0, so 3x1−2x2=153x_1 - 2x_2 = 15
Since x1>x2x_1 > x_2, assign x1=5x_1 = 5 and x2=0x_2 = 0. Calculate 3(5)−2(0)=153(5) - 2(0) = 15.

Key Concept

Solving quadratic equations by rearranging terms, expanding binomials, factoring out the greatest common factor, and solving for roots.
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Topics

Solving Quadratic Equations by Factoring

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