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Zorluk: OrtaRadical and Rational Equations
If xx satisfies the equation below, what is the value of x−1x - 1?
30−2x=x−3\sqrt{30 - 2x} = x - 3
  1. A
    -4
  2. B
    2
  3. 6Cevap
  4. D
    8

Cevap

6
The correct answer is 66. Squaring both sides of the equation 30−2x=x−3\sqrt{30 - 2x} = x - 3 yields 30−2x=x2−6x+930 - 2x = x^2 - 6x + 9. Rearranging terms into standard quadratic form gives x2−4x−21=0x^2 - 4x - 21 = 0, which factors as (x−7)(x+3)=0(x - 7)(x + 3) = 0. This gives potential solutions of x=7x = 7 and x=−3x = -3. Checking these in the original equation shows that x=7x = 7 is valid since 30−2(7)=7−3  ⟹  4=4\sqrt{30 - 2(7)} = 7 - 3 \implies 4 = 4, while x=−3x = -3 is extraneous since 30−2(−3)=−3−3  ⟹  6≠−6\sqrt{30 - 2(-3)} = -3 - 3 \implies 6 \neq -6. Therefore, the value of the requested expression x−1x - 1 is 7−1=67 - 1 = 6.

Adım Adım Çözüm

1
Square both sides of the equation to eliminate the radical.
30−2x=(x−3)230 - 2x = (x - 3)^2
To solve a radical equation, squaring both sides isolates the terms under the square root.
2
Expand the right side and move all terms to one side to set the quadratic equation to zero.
x2−4x−21=0x^2 - 4x - 21 = 0
Expanding (x−3)2(x - 3)^2 yields x2−6x+9x^2 - 6x + 9. Subtracting 3030 and adding 2x2x to both sides results in a standard quadratic form.
3
Factor the quadratic equation.
(x−7)(x+3)=0(x - 7)(x + 3) = 0
Factoring the quadratic helps find the potential solutions for xx.
4
Identify potential solutions and substitute them back into the original equation to check for extraneous solutions.
x=7x = 7 is the only valid solution; x=−3x = -3 is extraneous.
Substituting x=7x = 7 gives 30−14=7−3\sqrt{30 - 14} = 7 - 3, which simplifies to 4=44 = 4 (true). Substituting x=−3x = -3 gives 30−2(−3)=−3−3\sqrt{30 - 2(-3)} = -3 - 3, which simplifies to 6=−66 = -6 (false).
5
Calculate the value of the expression x−1x - 1 using the valid solution x=7x = 7.
7−1=67 - 1 = 6
The question asks for the value of x−1x - 1, so we substitute 77 for xx.

Anahtar Kavram

Solving radical equations and identifying extraneous solutions.
Tahmini Süre:1m 35s
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Konular

Radical and Rational Equations

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