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Difficulty: EasySolving Quadratic Equations by Factoring

If x(x−6)=7x(x - 6) = 7, what is the positive difference between the two solutions to this equation?

  1. A
    6
  2. 8Answer
  3. C
    10
  4. D
    14
  5. E
    0

Answer

The positive difference between the two solutions is 88.
First, expand the left side of the equation to get x2−6x=7x^2 - 6x = 7. Next, subtract 77 from both sides to write the equation in standard form: x2−6x−7=0x^2 - 6x - 7 = 0. Factoring this quadratic expression gives (x−7)(x+1)=0(x - 7)(x + 1) = 0. Setting each factor to zero yields the solutions x=7x = 7 and x=−1x = -1. The positive difference between these two solutions is 7−(−1)=87 - (-1) = 8.

Step-by-Step Solution

1
Distribute the xx on the left side of the equation.
x2−6x=7x^2 - 6x = 7
Multiplying xx by (x−6)(x - 6) expands the left side to prepare for standard form.
2
Subtract 77 from both sides to set the quadratic equation to zero.
x2−6x−7=0x^2 - 6x - 7 = 0
Setting the quadratic equation to standard form ax2+bx+c=0ax^2 + bx + c = 0 is necessary before factoring.
3
Factor the quadratic trinomial.
(x−7)(x+1)=0(x - 7)(x + 1) = 0
We need two numbers that multiply to −7-7 and add to −6-6, which are −7-7 and 11.
4
Set each factor to zero to find the solutions.
x=7x = 7 and x=−1x = -1
The zero product property states that if a product is zero, at least one factor must be zero.
5
Calculate the positive difference between the two solutions.
7−(−1)=87 - (-1) = 8
The positive difference is found by subtracting the smaller solution from the larger solution.

Key Concept

Solving Quadratic Equations by Factoring
Estimated Time:1m 0s
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Topics

Solving Quadratic Equations by Factoring

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