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Difficulty: MediumRadical and Rational Equations
In the equation below, xx is a real number.
7−6x=x+8\sqrt{7 - 6x} = x + 8
What is the value of x+5x + 5?
  1. A
    8
  2. B
    -14
  3. 2Answer
  4. D
    -3

Answer

The correct answer is the value 2.
Substituting the only valid solution of the radical equation, which is −3-3, into the expression x+5x + 5 yields 22.

Step-by-Step Solution

1
Square both sides of the equation to eliminate the radical sign.
7−6x=(x+8)27 - 6x = (x + 8)^2, which expands to 7−6x=x2+16x+647 - 6x = x^2 + 16x + 64.
Squaring both sides of a radical equation allows it to be rewritten as a standard polynomial equation.
2
Rearrange the terms to set the quadratic equation equal to zero.
x2+22x+57=0x^2 + 22x + 57 = 0
Grouping all terms on one side of the equation puts it in standard quadratic form.
3
Factor the quadratic equation to find potential solutions for xx.
(x+19)(x+3)=0(x + 19)(x + 3) = 0, which gives the potential solutions x=−19x = -19 and x=−3x = -3.
Factoring allows us to solve the quadratic equation easily.
4
Test the potential solutions in the original equation to identify any extraneous solutions.
Substituting x=−3x = -3 yields 7−6(−3)=−3+8⇒25=5\sqrt{7 - 6(-3)} = -3 + 8 \Rightarrow \sqrt{25} = 5, which is true. Substituting x=−19x = -19 yields 7−6(−19)=−19+8⇒121=−11\sqrt{7 - 6(-19)} = -19 + 8 \Rightarrow \sqrt{121} = -11, which is false because a principal square root must be non-negative. Therefore, x=−3x = -3 is the only valid solution.
Squaring both sides can introduce extraneous solutions that must be discarded.
5
Substitute the valid solution x=−3x = -3 into the target expression x+5x + 5.
−3+5=2-3 + 5 = 2
The question asks for the value of the expression x+5x + 5.

Key Concept

Solving radical equations by squaring and checking for extraneous solutions.
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Topics

Radical and Rational Equations

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