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Question

Difficulty: EasySolving Quadratic Equations by Factoring

What is the sum of the solutions to the quadratic equation x(x−6)=16x(x - 6) = 16?

  1. A
    -6
  2. B
    10
  3. 6Answer
  4. D
    12
  5. E
    38

Answer

The sum of the solutions is 6.
Rearranging the equation to standard form gives x2−6x−16=0x^2 - 6x - 16 = 0. Factoring the trinomial yields (x−8)(x+2)=0(x - 8)(x + 2) = 0. Solving for xx by setting each factor to zero gives x=8x = 8 and x=−2x = -2. The sum of these solutions is 8+(−2)=68 + (-2) = 6.

Step-by-Step Solution

1
Distribute the variable on the left side of the equation.
x2−6x=16x^2 - 6x = 16
To solve a quadratic equation, we must first expand all products to identify the quadratic terms.
2
Subtract 16 from both sides of the equation to write it in standard form.
x2−6x−16=0x^2 - 6x - 16 = 0
A quadratic equation must be set to zero before factoring.
3
Factor the quadratic trinomial.
(x−8)(x+2)=0(x - 8)(x + 2) = 0
We need to find two numbers that multiply to −16-16 and add to −6-6. These numbers are −8-8 and 22.
4
Set each factor to zero and solve for xx.
x=8x = 8 and x=−2x = -2
According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero.
5
Add the two solutions to find their sum.
8+(−2)=68 + (-2) = 6
The question asks for the sum of the solutions.

Key Concept

Solving quadratic equations by factoring after rearranging terms into standard form.
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Topics

Solving Quadratic Equations by Factoring

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