Solve for $x$ in the equation $x^2+4 x-4=8$
Answer (1)
Rewrite the equation in standard form: x 2 + 4 x − 12 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values a = 1 , b = 4 , and c = − 12 into the formula and simplify.
The solutions are x = 2 and x = − 6 , so the final answer is x = − 6 , x = 2 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 4 x − 4 = 8 . Our goal is to solve for x , meaning we want to find the values of x that satisfy this equation.
Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . To do this, we subtract 8 from both sides of the equation: x 2 + 4 x − 4 − 8 = 8 − 8 x 2 + 4 x − 12 = 0
Applying the Quadratic Formula Now we can solve the quadratic equation x 2 + 4 x − 12 = 0 . We can use the quadratic formula, which is given by: x = 2 a − b ± b 2 − 4 a c In our equation, a = 1 , b = 4 , and c = − 12 . Plugging these values into the quadratic formula, we get: x = 2 ( 1 ) − 4 ± 4 2 − 4 ( 1 ) ( − 12 ) x = 2 − 4 ± 16 + 48 x = 2 − 4 ± 64 x = 2 − 4 ± 8
Finding the Solutions Now we can find the two possible values for x :
x 1 = 2 − 4 + 8 = 2 4 = 2 x 2 = 2 − 4 − 8 = 2 − 12 = − 6 So the solutions are x = 2 and x = − 6 .
Final Answer Therefore, the solutions to the equation x 2 + 4 x − 4 = 8 are x = 2 and x = − 6 .
Examples
Imagine you are designing a rectangular garden where the length is x + 6 feet and the width is x − 2 feet. The area of the garden is 0 square feet. The equation x 2 + 4 x − 12 = 0 models this situation, where x represents a dimension. Solving this equation gives you the possible values for x , which helps you determine the actual dimensions of the garden. This problem demonstrates how quadratic equations can be used to model and solve real-world problems involving areas and dimensions.
