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Difficulty: MediumRadical and Rational Equations

If xx satisfies the equation below, what is the value of x−3x - 3?

x−2x+7=4x - \sqrt{2x + 7} = 4
  1. A
    -2
  2. 6Answer
  3. C
    9
  4. D
    -4

Answer

6
The correct answer is the value 66. Isolating the radical in x−2x+7=4x - \sqrt{2x + 7} = 4 gives x−4=2x+7x - 4 = \sqrt{2x + 7}. Squaring both sides yields x2−8x+16=2x+7x^2 - 8x + 16 = 2x + 7, which simplifies to the quadratic equation x2−10x+9=0x^2 - 10x + 9 = 0. Factoring this equation gives (x−9)(x−1)=0(x - 9)(x - 1) = 0, yielding potential solutions of x=9x = 9 and x=1x = 1. Substituting these potential solutions back into the original equation reveals that x=9x = 9 is a valid solution because 9−25=49 - \sqrt{25} = 4, whereas x=1x = 1 is extraneous because 1−9=−2≠41 - \sqrt{9} = -2 \neq 4. The value of the expression x−3x - 3 is 9−3=69 - 3 = 6.

Step-by-Step Solution

1
Isolate the radical term in the equation.
x−4=2x+7x - 4 = \sqrt{2x + 7}
Before squaring both sides of a radical equation, it is best to isolate the radical term on one side.
2
Square both sides of the equation to eliminate the radical.
(x−4)2=2x+7(x - 4)^2 = 2x + 7
Squaring a square root removes the radical, allowing us to solve the resulting algebraic equation.
3
Expand the squared binomial and move all terms to one side to set the quadratic equation to zero.
x2−10x+9=0x^2 - 10x + 9 = 0
(x−4)2(x - 4)^2 expands to x2−8x+16x^2 - 8x + 16. Subtracting 2x2x and 77 from both sides gives the standard quadratic form ax2+bx+c=0ax^2 + bx + c = 0.
4
Factor the quadratic equation.
(x−9)(x−1)=0(x - 9)(x - 1) = 0
Finding two numbers that multiply to 99 and add to −10-10 gives −9-9 and −1-1. Thus, the potential solutions are x=9x = 9 and x=1x = 1.
5
Substitute both potential solutions back into the original equation to check for extraneous solutions.
x=9x = 9 is valid, while x=1x = 1 is extraneous.
Checking x=9x = 9: 9−2(9)+7=9−5=49 - \sqrt{2(9)+7} = 9 - 5 = 4 (true). Checking x=1x = 1: 1−2(1)+7=1−3=−2≠41 - \sqrt{2(1)+7} = 1 - 3 = -2 \neq 4 (false). Thus, x=9x = 9 is the only real solution.
6
Calculate the value of the requested expression x−3x - 3.
9−3=69 - 3 = 6
Since the only valid solution is x=9x = 9, we substitute 99 into the expression x−3x - 3 to find the final answer.

Key Concept

Radical equations require isolating the radical, squaring both sides, solving the resulting equation, and checking for extraneous solutions that do not satisfy the original equation.
Estimated Time:1m 30s
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Topics

Radical and Rational Equations

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