Solve this equation.
(x)/(x-2) + (x-1)/(x+2) =-1
Ms.Sue could you help me?
Absolutely not, jimmy mic dimmy !
I was not successful at helping you in your previous guise.
I can change my name if I want. Can anybody else help me then?
Apparently not.
To solve the equation:
1. Start by finding a common denominator for the two fractions. The common denominator will be (x-2)(x+2), as it contains both (x-2) and (x+2).
2. Multiply each term in the equation by the common denominator to eliminate the fractions. This step involves cross-multiplying.
(x)(x+2) + (x-1)(x-2) = -1(x-2)(x+2)
Simplify the left side:
x(x+2) + (x-1)(x-2) = -1(x^2 - 4)
Distribute:
x^2 + 2x + (x^2 - 3x - 2) = -x^2 + 4
Simplify further:
x^2 + 2x + x^2 - 3x - 2 = -x^2 + 4
Combine like terms:
2x^2 - x - 2 = -x^2 + 4
3. Move all the terms to one side of the equation:
2x^2 - x - 2 + x^2 - 4 = 0
Simplify:
3x^2 - x - 6 = 0
4. Now we have a quadratic equation in the form of ax^2 + bx + c = 0. We can solve this either by factoring, completing the square, or by using the quadratic formula.
In this case, the quadratic equation cannot be easily factored, so we will use the quadratic formula:
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 3, b = -1, and c = -6:
x = (-(-1) ± √((-1)^2 - (4 * 3 * -6))) / (2 * 3)
Simplify:
x = (1 ± √(1 + 72)) / 6
x = (1 ± √73) / 6
There are two possible solutions:
x ≈ -1.29
x ≈ 1.29
Therefore, the solutions to the equation are x ≈ -1.29 and x ≈ 1.29.