How do you solve 2{x^2} - 5x - 12 = 0 by completing the square?

To solve the quadratic equation 2x^2 - 5x - 12 = 0 by completing the square, follow these steps:

Start with the equation in the form ax^2 + bx + c = 0, where a = 2, b = -5, and c = -12.

Divide the entire equation by the coefficient of the x^2 term (in this case, 2) to make the coefficient of x^2 equal to 1. This gives you:

x^2 - (5/2)x - 6 = 0

Move the constant term to the other side of the equation:
x^2 - (5/2)x = 6

To complete the square, we need to add and subtract a value that will make the left side of the equation a perfect square trinomial. To do this, take half of the coefficient of x (which is -5/2) and square it:
(-5/2)^2 = 25/4

Add and subtract 25/4 inside the parentheses:
x^2 - (5/2)x + 25/4 - 25/4 = 6

Rewrite the equation with the perfect square trinomial:
(x^2 - (5/2)x + 25/4) - 25/4 = 6

Simplify the equation:
(x - 5/2)^2 - 25/4 = 6

Add 25/4 to both sides of the equation to isolate the perfect square trinomial:
(x - 5/2)^2 = 6 + 25/4

Combine the constants on the right side:
(x - 5/2)^2 = 24/4 + 25/4

Simplify the right side further:
(x - 5/2)^2 = 49/4

Take the square root of both sides to solve for x:
x - 5/2 = ±√(49/4)

Simplify the right side:
x - 5/2 = ±(7/2)

Add 5/2 to both sides to isolate x:
x = 5/2 ± 7/2

Now, you have two solutions for x:

x = (5/2) + (7/2) = 12/2 = 6
x = (5/2) - (7/2) = -2/2 = -1
So, the solutions to the equation 2x^2 - 5x - 12 = 0 are x = 6 and x = -1.