Please show me how to solve this problem.
8/x^2-16 + 3/4 = 1/x-4
multiply each of the terms by x^2, collect your like terms and bring them all to the left side.
You will have a quadratic equation.
BTW, since you did not use brackets I will assume your last term is (1/x)-4
and not 1/(x-4), otherwise the solution process would change.
looking at your equation and the values again, you probably meant:
8/(x^2-16) + 3/4 = 1/(x-4)
in that case, notice that x^2-16 factors into (x+4)(x-4)
so multiply each term by 4(x+4)(x-4), which will clear all your denominators, and you will have a quadratic.
To solve the equation 8/(x^2-16) + 3/4 = 1/(x-4), we need to follow these steps:
Step 1: Factor the denominator x^2 - 16.
The difference of squares formula states that x^2 - 16 can be factored as (x + 4)(x - 4).
Step 2: Multiply each term by 4(x + 4)(x - 4).
Multiply every term in the equation by the LCD (least common denominator), which is 4(x + 4)(x - 4). This will clear all the denominators in the equation.
After multiplying, the equation becomes:
8 * 4(x + 4)(x - 4)/(x^2 - 16) + 3/4 * 4(x + 4)(x - 4) = 1 * 4(x + 4)(x - 4)/(x - 4)
Simplifying, we get:
32(x + 4)(x - 4) + 3(x + 4)(x - 4) = 4(x + 4)
Step 3: Simplify the equation.
Expand the terms using the distributive property.
32(x^2 - 16) + 3(x^2 - 16) = 4(x + 4)
32x^2 - 512 + 3x^2 - 48 = 4x + 16
Combine like terms:
35x^2 - 560 = 4x + 16
Step 4: Move all terms to one side of the equation.
Subtract 4x from both sides and add 560 to both sides to get:
35x^2 - 4x - 576 = 0
Now you have a quadratic equation: 35x^2 - 4x - 576 = 0.
To solve for x, you can either factor the quadratic equation or use the quadratic formula. However, the quadratic equation 35x^2 - 4x - 576 does not easily factor, so we will use the quadratic formula.
Step 5: Use the quadratic formula to find the value(s) of x.
The quadratic formula is x = (-b ± √(b^2 - 4ac))/(2a), where a, b, and c are the coefficients in the quadratic equation ax^2 + bx + c = 0.
For our equation 35x^2 - 4x - 576 = 0, we have a = 35, b = -4, and c = -576.
Plugging these values into the quadratic formula, we get:
x = (-(-4) ± √((-4)^2 - 4 * 35 * (-576)))/(2 * 35)
Simplifying further:
x = (4 ± √(16 + 80640))/70
x = (4 ± √80656)/70
Since the value under the square root cannot be simplified, we leave it as is.
Therefore, the solutions for x are:
x = (4 + √80656)/70
x = (4 - √80656)/70
These are the two possible solutions for the equation 8/(x^2-16) + 3/4 = 1/(x-4).