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Zorluk: OrtaQuadratic Equations and Factoring

If xx is a real number that satisfies the equation (x−3)2=16(x - 3)^2 = 16, which of the following could be the value of x2−4xx^2 - 4x? Select all such values.

  1. A
    −5-5
  2. 55Cevap
  3. 2121Cevap
  4. D
    4545
  5. E
    7777

Cevap

The possible values of x2−4xx^2 - 4x are 55 and 2121.
Taking the square root of both sides of (x−3)2=16(x - 3)^2 = 16 yields x−3=±4x - 3 = \pm 4. This gives two solutions for xx: x=7x = 7 and x=−1x = -1. Substituting x=7x = 7 into x2−4xx^2 - 4x gives 49−28=2149 - 28 = 21. Substituting x=−1x = -1 into x2−4xx^2 - 4x gives (−1)2−4(−1)=1+4=5(-1)^2 - 4(-1) = 1 + 4 = 5. Thus, both 55 and 2121 are valid values.

Adım Adım Çözüm

1
Solve the quadratic equation for xx
x−3=4  ⟹  x=7x - 3 = 4 \implies x = 7 or x−3=−4  ⟹  x=−1x - 3 = -4 \implies x = -1
Taking the square root of both sides of (x−3)2=16(x - 3)^2 = 16 yields both positive and negative roots.
2
Substitute the first root into the expression x2−4xx^2 - 4x
72−4(7)=49−28=217^2 - 4(7) = 49 - 28 = 21
Evaluating the target expression for x=7x = 7.
3
Substitute the second root into the expression x2−4xx^2 - 4x
(−1)2−4(−1)=1+4=5(-1)^2 - 4(-1) = 1 + 4 = 5
Evaluating the target expression for x=−1x = -1.

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Quadratic Equations and Factoring
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