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Difficulty: MediumSolving Quadratic Equations by Factoring

The sum of 99 and the product of a number and 66 less than that number is equal to 33 more than the number. What is the sum of all possible values of this number?

  1. 7Answer
  2. B
    1
  3. C
    5
  4. D
    -2
  5. E
    -7

Answer

7
The correct answer is 77. The verbal description translates to x(x−6)+9=x+3x(x - 6) + 9 = x + 3. Distributing the left side gives x2−6x+9=x+3x^2 - 6x + 9 = x + 3. Subtracting xx and 33 from both sides results in x2−7x+6=0x^2 - 7x + 6 = 0. Factoring the quadratic expression gives (x−6)(x−1)=0(x - 6)(x - 1) = 0, which yields the solutions x=6x = 6 and x=1x = 1. Summing these values gives 6+1=76 + 1 = 7.

Step-by-Step Solution

1
Translate the verbal statement into an algebraic equation.
x(x−6)+9=x+3x(x - 6) + 9 = x + 3, where xx represents the unknown number.
Establishing the relationship between the algebraic expressions defined by the problem.
2
Expand the left side of the equation and combine like terms to set the equation to zero.
x2−6x+9=x+3  ⟹  x2−7x+6=0x^2 - 6x + 9 = x + 3 \implies x^2 - 7x + 6 = 0.
Quadratic equations must be set to zero before they can be solved by factoring.
3
Factor the quadratic equation over the integers.
(x−6)(x−1)=0(x - 6)(x - 1) = 0.
Finding two numbers that multiply to 66 and add up to −7-7 allows us to factor the trinomial.
4
Apply the zero-product property to find the individual roots.
x=6x = 6 and x=1x = 1.
If the product of two factors is zero, at least one of the factors must be zero.
5
Sum the possible values of the number.
6+1=76 + 1 = 7.
The question asks for the sum of all possible values of the number.

Key Concept

Solving Quadratic Equations by Factoring
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Topics

Solving Quadratic Equations by Factoring

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