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Difficulty: MediumRational and Radical Expressions and Equations
Consider the system of equations below:
y=x+37y = \sqrt{x + 37}
y=x−5y = x - 5
If (x,y)(x, y) is a real solution to this system, what is the value of xyxy?
  1. A
    -12
  2. B
    -4
  3. C
    6
  4. D
    24
  5. 84Answer

Answer

84
The correct answer is 84 because solving the system yields the quadratic equation x2−11x−12=0x^2 - 11x - 12 = 0, which has roots x=12x = 12 and x=−1x = -1. Substituting these back into the original equations shows that only x=12x = 12 is valid (giving y=7y = 7), while x=−1x = -1 is extraneous (since 36=6≠−6\sqrt{36} = 6 \neq -6). The product of the coordinates of the valid solution is 12×7=8412 \times 7 = 84.

Step-by-Step Solution

1
Set the two expressions for yy equal to each other to solve for xx.
x+37=x−5\sqrt{x + 37} = x - 5
Since both equations are solved for yy, their right-hand sides must be equal at the point of intersection.
2
Square both sides of the equation to eliminate the radical.
x+37=(x−5)2⇒x+37=x2−10x+25x + 37 = (x - 5)^2 \Rightarrow x + 37 = x^2 - 10x + 25
Squaring is the inverse operation of the square root. The binomial on the right must be expanded fully using (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2.
3
Rearrange the quadratic equation into standard form.
x2−11x−12=0x^2 - 11x - 12 = 0
Subtracting xx and 3737 from both sides sets the quadratic equation to zero so it can be solved.
4
Factor the quadratic equation to find the algebraic solutions.
(x−12)(x+1)=0⇒x=12 or x=−1(x - 12)(x + 1) = 0 \Rightarrow x = 12 \text{ or } x = -1
Factoring the trinomial allows us to find the roots by setting each linear factor to zero.
5
Substitute each solution back into the original system to check for extraneous solutions.
For x=12x = 12: y=12−5=7y = 12 - 5 = 7 and y=12+37=7y = \sqrt{12 + 37} = 7 (Valid). For x=−1x = -1: y=−1−5=−6y = -1 - 5 = -6, but y=−1+37=6≠−6y = \sqrt{-1 + 37} = 6 \neq -6 (Extraneous).
Squaring both sides of an equation can introduce extraneous solutions that do not satisfy the original radical relationship because the principal square root must be non-negative.
6
Calculate the product xyxy of the coordinates of the valid solution.
xy=12×7=84xy = 12 \times 7 = 84
The question asks for the product of xx and yy for the real solution (12,7)(12, 7).

Key Concept

Solving systems containing radical equations and verifying for extraneous solutions.
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Topics

Rational and Radical Expressions and Equations

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