Solve: x^-2 - x^-1= 5/4
I have the answer as (5+/- sqrt29)/2
Is this right?
To solve the equation x^(-2) - x^(-1) = 5/4, we can follow these steps:
1. Start by multiplying the entire equation by x^2 to eliminate the negative exponents:
(x^(-2) * x^2) - (x^(-1) * x^2) = (5/4) * x^2
Simplifying this equation gives:
1 - x = (5/4) * x^2
2. Rearrange the equation to obtain a quadratic equation:
(5/4) * x^2 + x - 1 = 0
3. Multiply the entire equation by 4 to remove the fraction:
5x^2 + 4x - 4 = 0
4. Now we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
By comparing the quadratic equation to ax^2 + bx + c = 0, we have:
a = 5, b = 4, and c = -4
Substituting these values into the formula and simplifying gives:
x = (-4 ± √(4^2 - 4 * 5 * -4)) / (2 * 5)
= (-4 ± √(16 + 80)) / 10
= (-4 ± √96) / 10
= (-4 ± 4√6) / 10
= (1 ± √6) / 5
Hence, the correct answer is x = (1 ± √6) / 5, which can alternatively be expressed as (5 ± √29) / 10. Therefore, your answer of (5 ± √29) / 2 is not correct.