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, you can follow these steps:
Step 1: Simplify the equation by finding a common denominator for the terms on the left side. The common denominator for x^-2 and x^-1 is x^-2, since x^-1 = x^-2/x.
Now, the equation becomes x^-2 - (1/x)(x^-2) = 5/4.
Step 2: Combine the terms on the left side of the equation by subtracting their numerators.
You have x^-2 - (x^-2)/x = 5/4.
Step 3: Now, multiply both sides of the equation by x to get rid of the denominators.
You obtain x^-2 * x - (x^-2 * x)/x = (5/4) * x.
This simplifies to x^-1 - x^-2 = (5/4) * x.
Step 4: Rearrange the equation to one side and set it equal to zero.
You have x^-1 - x^-2 - (5/4) * x = 0.
Step 5: Now you can solve this equation using quadratic methods. Let's introduce a variable substitution. Let's set z = x^-1.
Substituting z into the equation, you get z - z^2 - (5/4) * (1/z) = 0.
Step 6: Multiply the entire equation by z to eliminate the denominators.
This results in z^2 - z^3 - (5/4) = 0.
Step 7: Rearrange the equation to form a quadratic equation.
You have z^3 - z^2 + (5/4) = 0.
Step 8: Solve the equation using quadratic methods.
One way to solve this is by using the quadratic formula.
The quadratic formula is given by z = (-b±√(b^2-4ac))/(2a).
Comparing your equation to the standard quadratic equation form (az^2 + bz + c = 0), we have a = 1, b = -1, and c = 5/4.
Using the quadratic formula, we can substitute these values in and solve for z.
z = (1±√((-1)^2 - 4(1)(5/4)))/(2(1)).
Simplifying this, we get z = (1±√(1 - 5/4))/2.
z = (1±√(4/4 - 5/4))/2.
z = (1±√(-1/4))/2.
Since the discriminant is negative, there are no real solutions for z.
Step 9: Substituting back the expression for z, which is z = x^-1, we can conclude that there are no real solutions for x in the original equation x^-2 - x^-1 = 5/4.
Thus, (5+/-√29)/2 is not the correct solution to the equation.