solve by factoring
6x - 5 = 6/x
Multiply both sides by x.
6x^2 - 5x = 6
6x^2 - 5x - 6 = 0
(2x - 3)(3x +2) = 0
To solve the equation 6x - 5 = 6/x by factoring, we need to try to rewrite the equation as a quadratic equation in terms of x.
First, let's move all the terms to the left side of the equation:
6x - 5 - 6/x = 0
To factorize the equation, we need to have a quadratic equation. We can achieve this by multiplying both sides of the equation by x:
(x)(6x - 5 - 6/x) = (x)(0)
6x^2 - 5x - 6 = 0
Now we have a quadratic equation in the standard form: ax^2 + bx + c = 0, where a = 6, b = -5, and c = -6.
Next, we need to factorize the quadratic equation. We are looking for two numbers that multiply to ac (product of a and c) and add up to b.
In this case, ac = 6 * -6 = -36, and b = -5. Let's consider the numbers that multiply to -36 and add up to -5:
-9 and 4
So, we can rewrite the quadratic equation as:
(2x - 3)(3x + 2) = 0
Now, we can solve for x by setting each factor equal to zero and solving for x:
2x - 3 = 0 or 3x + 2 = 0
For the first equation, adding 3 to both sides gives:
2x = 3
Dividing both sides by 2 gives:
x = 3/2 = 1.5
For the second equation, subtracting 2 from both sides gives:
3x = -2
Dividing both sides by 3 gives:
x = -2/3 ≈ -0.67
Therefore, the solutions to the equation 6x - 5 = 6/x are x = 1.5 and x ≈ -0.67.