Solve the equation by completing the square. If the solutions are real, give exact and approximate answers. Otherwise, list the exact solutions. 12x=-3x^2-14 Please help!!!!! :(
To solve the equation by completing the square, follow these steps:
Step 1: Move all the terms to one side of the equation to set it equal to zero.
-3x^2 - 12x - 14 = 0
Step 2: Divide the entire equation by the coefficient of the squared term to make it 1.
x^2 + 4x + 4.6667 = 0
Step 3: Complete the square by adding the square of half the coefficient of the x-term to both sides of the equation.
x^2 + 4x + 4 = 4.6667 + 4
Simplifying this equation gives:
(x + 2)^2 = 8.6667
Step 4: Take the square root of both sides of the equation.
√[(x + 2)^2] = √8.6667
Simplifying further:
x + 2 = ±√8.6667
Step 5: Solve for x by isolating it on one side of the equation.
For positive square root:
x = -2 + √8.6667
For negative square root:
x = -2 - √8.6667
These are the exact solutions to the equation. If you need approximate answers, you can calculate the square root of 8.6667 and simplify the expressions further using a calculator.
3x^2 + 12x = -14
x^2 + 4x = -14/3
x^2 + 4x + 4 = -2/3
(x + 2)^2 = -2/3