Solve the equation by completing the square. Round to the nearest hundredth if necessary.
x^2 - 3x - 12 = 0
To solve the equation by completing the square, we start by moving the constant term to the right side of the equation:
x^2 - 3x = 12
Next, we need to determine the value to add on both sides of the equation to complete the square. To do this, we take half of the coefficient of x, square it, and add it to both sides. In this case, half of the coefficient of x is (3/2) and squaring it gives us (3/2)^2 = 9/4.
Adding 9/4 on both sides of the equation:
x^2 - 3x + 9/4 = 12 + 9/4
(x - 3/2)^2 = 48/4 + 9/4
(x - 3/2)^2 = 57/4
Now, we take the square root of both sides of the equation to isolate x:
x - 3/2 = ±√(57/4)
x - 3/2 = ±(√57)/2
To solve for x, we separate into two cases:
1. x - 3/2 = (√57)/2
x = (√57)/2 + 3/2
2. x - 3/2 = -(√57)/2
x = -(√57)/2 + 3/2
Therefore, the solutions to the equation x^2 - 3x - 12 = 0 by completing the square are:
x ≈ 4.48 or x ≈ -1.48