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Difficulty: MediumQuadratic Equations and Factoring

If xx is a negative real number such that (x+5)2=36(x + 5)^2 = 36, what is the value of x2−2xx^2 - 2x?

  1. A
    −1-1
  2. B
    33
  3. C
    −99-99
  4. 143143Answer
  5. E
    9999

Answer

The value of x2−2xx^2 - 2x is 143143.
Taking the square root of both sides of (x+5)2=36(x + 5)^2 = 36 gives x+5=6x + 5 = 6 or x+5=−6x + 5 = -6. Solving for xx yields x=1x = 1 or x=−11x = -11. Because xx is given as a negative real number, x=−11x = -11. Evaluating the target expression at x=−11x = -11 yields (−11)2−2(−11)=121+22=143(-11)^2 - 2(-11) = 121 + 22 = 143.

Step-by-Step Solution

1
Solve the quadratic equation for xx by applying the square root property.
x+5=±36=±6x + 5 = \pm \sqrt{36} = \pm 6
Taking the square root of both sides yields both positive and negative root options.
2
Determine the two potential solutions for xx.
x=6−5=1x = 6 - 5 = 1 or x=−6−5=−11x = -6 - 5 = -11
Isolating xx gives two real solutions.
3
Select the valid solution based on the problem constraints.
x=−11x = -11
The stem specifies that xx must be a negative real number, so x=1x = 1 is rejected.
4
Substitute x=−11x = -11 into the expression x2−2xx^2 - 2x.
(−11)2−2(−11)=121+22=143(-11)^2 - 2(-11) = 121 + 22 = 143
Squaring a negative number yields a positive result, and subtracting a negative quantity adds to the total.

Key Concept

Solving quadratic equations using square roots and evaluating algebraic expressions
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Topics

Quadratic Equations and Factoring

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