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Zorluk: ZorRational and Radical Expressions and Equations

Which of the following is the complete set of real solutions to the equation 5x−9=x−3\sqrt{5x - 9} = x - 3?

  1. {9}\{9\}Cevap
  2. B
    {2,9}\{2, 9\}
  3. C
    {0,5}\{0, 5\}
  4. D
    {1.5,2}\{1.5, 2\}
  5. E
    {2}\{2\}

Cevap

{9}\{9\}
The correct option is the set containing only the value 9. To solve the equation, we square both sides to get 5x−9=x2−6x+95x - 9 = x^2 - 6x + 9, which simplifies to the quadratic equation x2−11x+18=0x^2 - 11x + 18 = 0. Factoring this yields the potential solutions 9 and 2. Substituting these back into the original equation, we find that 9 satisfies the equation while 2 results in a contradiction, making it extraneous.

Adım Adım Çözüm

1
Isolate the radical and square both sides of the equation.
5x−9=(x−3)25x - 9 = (x - 3)^2
Squaring both sides eliminates the square root to allow solving for the variable.
2
Expand the right-hand side using the binomial squaring rule.
5x−9=x2−6x+95x - 9 = x^2 - 6x + 9
The square of a binomial (a−b)2(a - b)^2 is a2−2ab+b2a^2 - 2ab + b^2.
3
Move all terms to one side to set the quadratic equation to zero, then factor.
x2−11x+18=0⇒(x−9)(x−2)=0x^2 - 11x + 18 = 0 \Rightarrow (x - 9)(x - 2) = 0
Rearranging terms simplifies the equation into a standard quadratic form that can be factored.
4
Solve for the potential roots and check for extraneous solutions in the original equation.
x=9x = 9 (valid) and x=2x = 2 (extraneous)
Checking x=9x = 9 gives 5(9)−9=9−3⇒6=6\sqrt{5(9) - 9} = 9 - 3 \Rightarrow 6 = 6 (true). Checking x=2x = 2 gives 5(2)−9=2−3⇒1=−1\sqrt{5(2) - 9} = 2 - 3 \Rightarrow 1 = -1 (false).

Anahtar Kavram

Solving radical equations and verifying for extraneous solutions
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Rational and Radical Expressions and Equations

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