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Difficulty: MediumSolving Quadratic Equations by Factoring

If (x−4)2=−4x+28(x - 4)^2 = -4x + 28, what is the sum of the solutions to the equation?

  1. A
    -8
  2. B
    -4
  3. 4Answer
  4. D
    8
  5. E
    28

Answer

The sum of the solutions to the equation is 44.
The correct answer of 44 is found by first expanding (x−4)2(x - 4)^2 to obtain x2−8x+16=−4x+28x^2 - 8x + 16 = -4x + 28. Setting the equation to zero gives the quadratic equation x2−4x−12=0x^2 - 4x - 12 = 0. Factoring the trinomial yields (x−6)(x+2)=0(x - 6)(x + 2) = 0, which results in the solutions x=6x = 6 and x=−2x = -2. Adding these solutions together gives 6+(−2)=46 + (-2) = 4.

Step-by-Step Solution

1
Expand the squared binomial on the left side of the equation.
x2−8x+16=−4x+28x^2 - 8x + 16 = -4x + 28
Applying the binomial expansion formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to (x−4)2(x - 4)^2 yields x2−8x+16x^2 - 8x + 16.
2
Rearrange the equation into standard quadratic form by adding 4x4x and subtracting 2828 from both sides.
x2−4x−12=0x^2 - 4x - 12 = 0
To solve a quadratic equation by factoring, all terms must be moved to one side so that the equation is set to zero.
3
Factor the quadratic expression x2−4x−12x^2 - 4x - 12.
(x−6)(x+2)=0(x - 6)(x + 2) = 0
Find two integers that multiply to the constant term −12-12 and add to the linear coefficient −4-4. These numbers are −6-6 and 22.
4
Set each binomial factor to zero and solve for xx.
x=6x = 6 or x=−2x = -2
By the zero-product property, if the product of two factors is zero, at least one of the factors must be zero.
5
Find the sum of the solutions.
6+(−2)=46 + (-2) = 4
Adding the two individual solutions gives the requested sum.

Key Concept

Solving quadratic equations by expanding binomials, rearranging terms into standard form, factoring the trinomial, and applying the zero-product property.

Alternative Method

Once the equation is rewritten in the standard form x2−4x−12=0x^2 - 4x - 12 = 0, Vieta's formulas can be used to find the sum of the solutions directly. For any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0, the sum of the roots is given by −ba-\frac{b}{a}. Substituting a=1a = 1 and b=−4b = -4 into the formula gives −−41=4-\frac{-4}{1} = 4.
Estimated Time:1m 0s
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Solving Quadratic Equations by Factoring

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