solve the ration equation: 2x/x-4 - 2x-5/x^2-10x+24= -3/x-6
Maybe could you mean:
2x / (x-4) - (2x-5) / (x^2-10x+24) = -3/ (x-6 )
now luckily 24 = 4 * 6 so we know what to do
2x / (x-4) - (2x-5) / [ (x-4)(x-6) ] = -3/ (x-6 )
first term by (x -6) / (x-6) etc so they are all (x-4)(x-6) on the bottom
2x(x-6) - (2x-5) = -3 (x-4)
2 x^2 - 12 x - 2 x + 5 = -3 x + 12
2 x^2 - 11 x - 7 = 0
I get 6.07 and -0.576
To solve the rational equation 2x/(x-4) - (2x-5)/(x^2-10x+24) = -3/(x-6), we can follow these steps:
Step 1: Simplify the denominators
The first denominator, x-4, is already simplified.
The second denominator, x^2-10x+24, factors to (x-4)(x-6).
Step 2: Find the common denominator
To add or subtract fractions, we need a common denominator. In this case, the common denominator is (x-4)(x-6).
Step 3: Multiply each term by the common denominator
Multiplying each term by the common denominator, we get:
2x(x-6) - (2x-5)(x-4) = -3(x-4)(x-6)
Step 4: Expand and simplify
Expanding and simplifying, we have:
2x^2 - 12x - (2x^2 - 13x + 20) = -3x^2 + 30x - 72
Simplifying further, we get:
2x^2 - 12x - 2x^2 + 13x - 20 = -3x^2 + 30x - 72
Combining like terms:
(-12x + 13x) + (2x^2 - 2x^2) - 20 = -3x^2 + 30x - 72
Simplifying:
x - 20 = -3x^2 + 30x - 72
Step 5: Move all terms to one side
Rearranging the equation, we have:
3x^2 + 29x - 52 = 0
Step 6: Solve the quadratic equation
We can solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, the quadratic equation cannot be easily factored, so we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the values a = 3, b = 29, and c = -52, we can calculate the solutions.
x = (-29 ± √(29^2 - 4 * 3 * -52)) / (2 * 3)
x = (-29 ± √(841 + 624)) / 6
x = (-29 ± √(1465)) / 6
Step 7: Simplify the square root
Since 1465 is not a perfect square, we can approximate the square root as follows:
x = (-29 ± √(1465)) / 6
x ≈ (-29 ± 38.27) / 6
So, the two possible solutions are:
x1 ≈ (-29 + 38.27) / 6 ≈ 1.55
x2 ≈ (-29 - 38.27) / 6 ≈ -11.21
Therefore, the solutions to the rational equation are x ≈ 1.55 and x ≈ -11.21.