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Difficulty: Very hardSolving Quadratic Equations by Factoring

A certain real number xx satisfies the condition that the square of 33 less than twice xx is equal to 88 times the quantity 33 minus xx. What is the ratio of the larger solution to the smaller solution of this equation?

  1. A
    −154-\frac{15}{4}
  2. −53-\frac{5}{3}Answer
  3. C
    −35-\frac{3}{5}
  4. D
    53\frac{5}{3}
  5. E
    33

Answer

The ratio of the larger solution to the smaller solution is −53-\frac{5}{3}.
The correct answer is −53-\frac{5}{3}. The verbal statement translates directly to (2x−3)2=8(3−x)(2x - 3)^2 = 8(3 - x). Expanding both sides yields 4x2−12x+9=24−8x4x^2 - 12x + 9 = 24 - 8x. Rearranging into standard form gives 4x2−4x−15=04x^2 - 4x - 15 = 0. Factoring by grouping yields (2x+3)(2x−5)=0(2x + 3)(2x - 5) = 0, giving the solutions x=−32x = -\frac{3}{2} and x=52x = \frac{5}{2}. The ratio of the larger root to the smaller root is 5/2−3/2=−53\frac{5/2}{-3/2} = -\frac{5}{3}.

Step-by-Step Solution

1
Translate the verbal description into an algebraic equation.
(2x−3)2=8(3−x)(2x - 3)^2 = 8(3 - x)
'Twice xx' is 2x2x, '3 less than twice xx' is 2x−32x - 3, and its square is (2x−3)2(2x - 3)^2. This is equal to 8 times the quantity 3−x3 - x.
2
Expand both sides of the equation.
4x2−12x+9=24−8x4x^2 - 12x + 9 = 24 - 8x
Expanding the binomial (2x−3)2(2x - 3)^2 gives 4x2−12x+94x^2 - 12x + 9 and distributing the right side gives 24−8x24 - 8x.
3
Rearrange the equation to set it equal to zero.
4x2−4x−15=04x^2 - 4x - 15 = 0
Add 8x8x and subtract 2424 from both sides to gather all terms on one side of the equation.
4
Factor the quadratic equation by grouping.
(2x+3)(2x−5)=0(2x + 3)(2x - 5) = 0
Find two numbers that multiply to 4×(−15)=−604 \times (-15) = -60 and add to −4-4. These numbers are −10-10 and 66. Rewrite the middle term as −10x+6x-10x + 6x and factor: 2x(2x−5)+3(2x−5)=0  ⟹  (2x+3)(2x−5)=02x(2x - 5) + 3(2x - 5) = 0 \implies (2x + 3)(2x - 5) = 0.
5
Solve for the roots of the equation.
x=−32x = -\frac{3}{2} and x=52x = \frac{5}{2}
Set each factor equal to zero: 2x+3=0  ⟹  x=−322x + 3 = 0 \implies x = -\frac{3}{2} and 2x−5=0  ⟹  x=522x - 5 = 0 \implies x = \frac{5}{2}.
6
Identify the larger and smaller solutions and compute their ratio.
−53-\frac{5}{3}
The larger solution is 52\frac{5}{2} and the smaller solution is −32-\frac{3}{2}. Their ratio is 5/2−3/2=−53\frac{5/2}{-3/2} = -\frac{5}{3}.

Key Concept

Solving quadratic equations of the form ax2+bx+c=0ax^2 + bx + c = 0 by factoring over the integers.
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Topics

Solving Quadratic Equations by Factoring

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