Solve for x. 4x(x-1)-5x(x)=3

4x^2 - 4x - 5x^2 = 3

x^2 + 4x + 3 = 0
(x+1)(x+3) = 0

x = -1 or -3

Thank you.

To solve for x in the equation 4x(x-1) - 5x(x) = 3, we need to simplify the equation and then use algebraic techniques to isolate the variable.

Step 1: Distribute the terms within the parentheses.
4x^2 - 4x - 5x^2 = 3

Step 2: Combine like terms on the left-hand side.
-4x^2 - 4x - 5x^2 = 3

Step 3: Combine the x^2 terms and the x terms.
-9x^2 - 4x = 3

Step 4: Move the constant term to the right-hand side to set the equation equal to zero.
-9x^2 - 4x - 3 = 0

Now, to solve for x, we can use either factoring, completing the square, or the quadratic formula. In this case, the quadratic formula seems to be the most suitable approach.

Step 5: Apply the quadratic formula, where ax^2 + bx + c = 0.
The quadratic formula is x = (-b ± √(b^2 - 4ac)) / (2a).

In our equation -9x^2 - 4x - 3 = 0, we have a = -9, b = -4, and c = -3. Plugging these values into the quadratic formula, we get:
x = (-(-4) ± √((-4)^2 - 4(-9)(-3))) / (2(-9))
x = (4 ± √(16 - 108)) / (-18)
x = (4 ± √(-92)) / (-18)

At this point, we realize that the term inside the square root results in a negative value, which means there are no real solutions to this equation. Therefore, the equation 4x(x-1) - 5x(x) = 3 has no real solutions for x.