Note: Enter your answer and show all the steps that you use to solve this problem in the space provided.
Solve the following equation. Show all your work.
x
x
−
2
+
x
−
1
x
+
1
=
−
1
To solve the equation:
1. Simplify the left side by finding a common denominator:
x(x-1) - 2(x+1) = -1(x+1)(x-1)
x^2 - x - 2x - 2 = -x^2 + x + x - 1
2x^2 - 4 = 0
2. Solve for x by factoring:
2(x^2 - 2) = 0
x^2 - 2 = 0
(x + √2)(x - √2) = 0
x = √2 or x = -√2
Therefore, the solution set is {√2, -√2}.
To solve the equation, follow these steps:
Step 1: Multiply both sides of the equation by (x + 1) to get rid of the denominator:
(x^2 - 2) + (x - 1) = -1 * (x + 1)
Step 2: Simplify both sides of the equation:
x^2 - 2 + x - 1 = -x - 1
Step 3: Combine like terms on the left side of the equation:
x^2 + x - 3 = -x - 1
Step 4: Move all terms to the left side of the equation:
x^2 + x + x + 3 + 1 = 0
Step 5: Combine like terms:
x^2 + 2x + 4 = 0
Step 6: Since the equation is a quadratic, try factoring.
Unfortunately, x^2 + 2x + 4 cannot be factored.
Step 7: Use the quadratic formula to find the solutions for x:
x = (-b ± √(b^2 - 4ac)) / 2a
For the equation x^2 + 2x + 4 = 0:
a = 1, b = 2, and c = 4.
Plugging in the values into the quadratic formula:
x = (-2 ± √(2^2 - 4(1)(4))) / (2(1))
x = (-2 ± √(4 - 16)) / 2
x = (-2 ± √(-12)) / 2
Step 8: Simplify the square root:
√(-12) = √(-1 * 4 * 3) = 2i√3
Step 9: Substitute the square root back into the equation:
x = (-2 ± 2i√3) / 2
Step 10: Simplify further:
x = -1 ± i√3
Therefore, the solutions for the equation x^2 - 2 + x - 1 = -x - 1 are:
x = -1 + i√3 and x = -1 - i√3.