Examkin

Soru

Zorluk: ZorRadical and Rational Equations

If xx is a solution to the equation 2x−3x−1−2x−2=1x2−3x+2\frac{2x - 3}{x - 1} - \frac{2}{x - 2} = \frac{1}{x^2 - 3x + 2}, what is the value of 2x−12x - 1?

  1. A
    1
  2. B
    5
  3. 6Cevap
  4. D
    9

Cevap

6
The correct answer is the value obtained by solving the rational equation for xx and then calculating the expression 2x−12x-1. Multiplying the equation by the least common denominator (x−1)(x−2)(x-1)(x-2) yields the quadratic equation 2x2−9x+7=02x^2 - 9x + 7 = 0, which factors into (2x−7)(x−1)=0(2x - 7)(x - 1) = 0. The potential solution x=1x = 1 is extraneous because it makes the denominators of the original equation equal to zero. Thus, the only valid solution is x=7/2x = 7/2. Substituting this value into the expression gives 2(7/2)−1=62(7/2) - 1 = 6.

Adım Adım Çözüm

1
Identify the least common denominator (LCD) of the rational expressions in the equation.
The LCD is (x−1)(x−2)=x2−3x+2(x - 1)(x - 2) = x^2 - 3x + 2. Since these denominators cannot be zero, we must have x≠1x \neq 1 and x≠2x \neq 2.
Finding the LCD allows us to eliminate the denominators by multiplying both sides of the equation.
2
Multiply both sides of the equation by the LCD, (x−1)(x−2)(x - 1)(x - 2), to clear the fractions.
(2x−3)(x−2)−2(x−1)=1(2x - 3)(x - 2) - 2(x - 1) = 1
This simplifies the rational equation into a polynomial equation.
3
Expand the products and simplify the equation.
(2x2−7x+6)−(2x−2)=1  ⟹  2x2−9x+8=1(2x^2 - 7x + 6) - (2x - 2) = 1 \implies 2x^2 - 9x + 8 = 1
Expanding the terms is necessary to group like terms and form a quadratic equation.
4
Subtract 1 from both sides to write the quadratic equation in standard form.
2x2−9x+7=02x^2 - 9x + 7 = 0
Standard form (ax2+bx+c=0ax^2 + bx + c = 0) is required to factor or apply the quadratic formula.
5
Factor the quadratic equation.
(2x−7)(x−1)=0(2x - 7)(x - 1) = 0, which gives potential solutions x=72x = \frac{7}{2} and x=1x = 1.
Factoring helps find the values of xx that satisfy the polynomial equation.
6
Check the potential solutions against the original denominators to identify any extraneous solutions.
For x=1x = 1, the denominators x−1x - 1 and x2−3x+2x^2 - 3x + 2 become zero, so x=1x = 1 is extraneous. For x=72x = \frac{7}{2}, the denominators are non-zero, so x=72x = \frac{7}{2} is the only valid solution.
Multiplying by variables can introduce extraneous solutions that make the original rational expressions undefined.
7
Calculate the value of the requested expression, 2x−12x - 1, using the valid solution x=72x = \frac{7}{2}.
2(72)−1=7−1=62\left(\frac{7}{2}\right) - 1 = 7 - 1 = 6
The question asks for the value of the expression 2x−12x - 1, not just the value of xx.

Anahtar Kavram

Solving rational equations by clearing denominators, factoring the resulting quadratic equation, and checking for extraneous solutions.
Bu soruyu puanla

Konular

Radical and Rational Equations

Bu konudan daha fazla soru