solve for
10/x^2-2x + 4/x = 5/x-2
3x/x+1 + 6/2x = 7/x
8(x-1)/x^2-4 = 4/x-2
check for any extraneous solutions! thanks!
I'll do one, and you try the others. If you get stuck, come on back.
10/(x^2-2x) + 4/x = 5/(x-2)
x^2-2x = x(x-2), so that's the LCD. Multiply through to get
10 + 4(x-2) = 5x
10 + 4x - 8 = 5x
x = 2
Unfortunately, the original equation has (x-2) in the denominator, so it is undefined there. No solution.
To solve these equations, we will follow these steps:
1. Simplify the equations if possible.
2. Find the common denominator.
3. Multiply both sides of the equation by the common denominator.
4. Simplify and solve the equation.
5. Check for any extraneous solutions.
Let's solve each equation step by step:
Equation 1: 10/x^2 - 2x + 4/x = 5/x - 2
1. Simplify: There isn't much to simplify here.
2. Find the common denominator: The common denominator is x^2.
3. Multiply both sides by the common denominator:
10 - 2x * x^2 + 4 * x = 5 * x^2 - 2 * x^2
4. Simplify and solve:
10x - 2x^3 + 4x = 5x^2 - 2x^2
-2x^3 + 14x - 8x^2 = 0
Rearrange to put it in the form of a quadratic equation:
-2x^3 - 8x^2 + 14x = 0
Factor out x:
x * (-2x^2 - 8x + 14) = 0
The solutions are x = 0 and the solutions of the quadratic equation:
-2x^2 - 8x + 14 = 0
5. Check for extraneous solutions: We'll check the original equation by substituting each solution into the equation. If the equation holds true, there are no extraneous solutions. If it doesn't hold true, we'll discard those solutions.
Let's move on to the next equation.
Equation 2: 3x/(x + 1) + 6/2x = 7/x
1. Simplify: Divide 6 by 2, resulting in 3/x.
2. Find the common denominator: The common denominator is x(x+1).
3. Multiply both sides by the common denominator:
3x * x(x+1)/(x+1) + 3 * (x+1)/(x+1) = 7 * x(x+1)/x
Simplify the equation:
3x^2 + 3(x+1) = 7(x(x+1))/x
Simplify further:
3x^2 + 3x + 3 = 7(x^2 + x)/x
4. Simplify and solve:
Distribute on both sides:
3x^2 + 3x + 3 = 7x + 7
Rearrange to put it in the form of a quadratic equation:
3x^2 - 4x + 4 = 0
Use the quadratic formula to solve for x:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 3, b = -4, and c = 4.
Evaluating the formula, we get:
x ≈ 1.28 and x ≈ 0.22
5. Check for extraneous solutions: Substitute each solution into the original equation and verify whether it holds true. If it doesn't, discard those solutions.
Now, let's solve the last equation.
Equation 3: 8(x-1)/x^2 - 4 = 4/x - 2
1. Simplify: There is not much simplification required here.
2. Find the common denominator: The common denominator is x^2.
3. Multiply both sides by the common denominator:
8(x-1) - 4 * x^2 = 4 - 2 * x^2
4. Simplify and solve:
Expand both sides:
8x - 8 - 4x^2 = 4 - 2x^2
Move all terms to one side:
6x^2 + 8x - 12 = 0
Divide through by 2:
3x^2 + 4x - 6 = 0
Use the quadratic formula to solve for x:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
In this case, a = 3, b = 4, and c = -6.
Evaluating the formula, we get:
x ≈ 0.92 and x ≈ -1.92
5. Check for extraneous solutions: Substitute each solution into the original equation and verify whether it holds true. If it doesn't, discard those solutions.
Remember to verify the solutions obtained by substituting them back into the original equation.