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Difficulty: MediumRadical and Rational Equations

If xx is a real solution to the equation 2x+7−x=2\sqrt{2x + 7} - x = 2, what is the value of x+4x + 4?

  1. 5Answer
  2. B
    1
  3. C
    7
  4. D
    3

Answer

5
The correct answer is 5. To solve the equation 2x+7−x=2\sqrt{2x + 7} - x = 2, we first isolate the radical to get 2x+7=x+2\sqrt{2x + 7} = x + 2. Squaring both sides yields 2x+7=x2+4x+42x + 7 = x^2 + 4x + 4. Rearranging this equation into standard quadratic form gives x2+2x−3=0x^2 + 2x - 3 = 0, which factors as (x+3)(x−1)=0(x + 3)(x - 1) = 0. This gives two potential solutions: x=1x = 1 and x=−3x = -3. Checking these in the original equation shows that x=1x = 1 is valid, whereas x=−3x = -3 is extraneous because 2(−3)+7−(−3)=1+3=4≠2\sqrt{2(-3)+7} - (-3) = 1 + 3 = 4 \neq 2. Therefore, the only real solution is x=1x = 1. Substituting this into the expression x+4x + 4 gives 1+4=51 + 4 = 5.

Step-by-Step Solution

1
Isolate the radical expression on one side of the equation.
2x+7=x+2\sqrt{2x + 7} = x + 2
This sets up the equation to eliminate the radical by squaring both sides.
2
Square both sides of the equation to eliminate the radical.
2x+7=(x+2)22x + 7 = (x + 2)^2
Squaring a square root removes the radical, allowing the equation to be solved algebraically.
3
Expand the squared binomial on the right side.
2x+7=x2+4x+42x + 7 = x^2 + 4x + 4
Expanding (x+2)2(x+2)^2 yields a quadratic term, a linear term, and a constant term.
4
Rearrange the equation to set it equal to zero.
x2+2x−3=0x^2 + 2x - 3 = 0
Subtracting 2x2x and 77 from both sides simplifies the equation into standard quadratic form.
5
Factor the quadratic equation.
(x+3)(x−1)=0(x + 3)(x - 1) = 0
Factoring allows us to find the potential solutions for xx.
6
Identify potential solutions and check for extraneous solutions by substituting them back into the original equation.
For x=1x = 1, 2(1)+7−1=3−1=2\sqrt{2(1) + 7} - 1 = 3 - 1 = 2, which is true. For x=−3x = -3, 2(−3)+7−(−3)=1+3=4\sqrt{2(-3) + 7} - (-3) = 1 + 3 = 4, which does not equal 22.
Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical equation. Thus, x=−3x = -3 is extraneous, leaving x=1x = 1 as the only valid solution.
7
Calculate the value of the expression x+4x + 4 using the valid solution.
1+4=51 + 4 = 5
The question asks for the value of x+4x + 4, so we substitute the valid solution x=1x = 1 into this expression.

Key Concept

Solving radical equations and checking for extraneous solutions.
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Topics

Radical and Rational Equations

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