solve: 1 / x + 2 + 1 / x - 2= 1 / x^2 - 4
a)-2
b) 2
c)-1/2
d) 1/2
you probably meant to type
1/(x + 2) + 1/(x - 2) = 1/(x^2 - 4)
multiply each term by x^2 - 4, ( which factors to (x+2)(x-2) )
x-2 + x+2 = 1
2x = 1
x = 1/2
multiply both sides by (x+2)(x-2)
x-2+x+2=1
x= 1/2
thanks reiny
To solve the given equation, we need to combine like terms and remove any fractions. Let's go step by step:
1. Start by multiplying all terms of the equation by the denominator of the fractions, which is (x + 2)(x - 2) or (x^2 - 4):
(x + 2)(x - 2) * (1 / x + 2) + (x + 2)(x - 2) * (1 / x - 2) = (x + 2)(x - 2) * (1 / x^2 - 4)
2. Simplify each term by distributing:
(x - 2) + (x + 2) = (1 / x^2 - 4)(x^2 - 4)
(x - 2 + x + 2) = (1 / x^2 - 4)(x^2 - 4)
(2x) = (1 / x^2 - 4)(x^2 - 4)
3. Distribute (1 / x^2 - 4) to (x^2 - 4):
(2x) = (1)(x^2 - 4) - (4)(x^2 - 4)
2x = x^2 - 4 - 4x^2 + 16
4. Combine like terms:
2x = -3x^2 + 12
5. Move all terms to one side of the equation:
0 = -3x^2 + 12 - 2x
0 = -3x^2 - 2x + 12
6. Rearrange the equation to the standard quadratic form, ax^2 + bx + c = 0:
3x^2 + 2x - 12 = 0
Now, we have a quadratic equation. We can either factor it or use the quadratic formula to find the roots.
Factoring method:
1. Multiply the coefficient of the x^2 term (3) with the constant term (-12):
-12 * 3 = -36
2. Look for two numbers that multiply to -36 and add up to the coefficient of the x term (2):
The numbers are 6 and -6.
3. Rewrite the quadratic equation by splitting the x term:
3x^2 + 6x - 6x - 12 = 0
4. Group the terms and factor by grouping:
(3x^2 + 6x) - (6x + 12) = 0
3x(x + 2) - 6(x + 2) = 0
5. Factor out the common term (x + 2):
(x + 2)(3x - 6) = 0
6. Set each factor equal to zero and solve for x:
x + 2 = 0 (Factor 1)
3x - 6 = 0 (Factor 2)
Solving Factor 1:
x = -2
Solving Factor 2:
3x = 6
x = 2
Therefore, the possible solutions are x = -2 and x = 2.
Comparing the solutions to the given options:
a) -2: Matches one of the solutions.
b) 2: Matches one of the solutions.
c) -1/2: No intersection with the solutions.
d) 1/2: No intersection with the solutions.
So, the correct solutions are a) -2 and b) 2.