Solve for x in the equation x²+4x-4 = 8.
Answer (1)
Rewrite the equation x 2 + 4 x − 4 = 8 as x 2 + 4 x = 12 .
Complete the square by adding ( 2 4 ) 2 = 4 to both sides: x 2 + 4 x + 4 = 16 .
Rewrite as a squared term: ( x + 2 ) 2 = 16 , then take the square root: x + 2 = ± 4 .
Solve for x : x = − 2 ± 4 , which gives x = 2 or x = − 6 . The final answer is x = − 6 or x = 2 .
Explanation
Understanding the Problem We are given the equation x 2 + 4 x − 4 = 8 and asked to solve for x using the method of completing the square.
Isolating the x Terms First, we rewrite the equation by adding 4 to both sides to isolate the terms with x on the left side: x 2 + 4 x = 8 + 4 x 2 + 4 x = 12
Completing the Square To complete the square, we need to add a value to both sides of the equation such that the left side becomes a perfect square trinomial. We take half of the coefficient of the x term, which is 2 4 = 2 , and square it: 2 2 = 4 . So, we add 4 to both sides of the equation: x 2 + 4 x + 4 = 12 + 4 x 2 + 4 x + 4 = 16
Rewriting as a Squared Term Now, we can rewrite the left side as a squared term: ( x + 2 ) 2 = 16
Taking the Square Root Next, we take the square root of both sides of the equation: ( x + 2 ) 2 = ± 16 x + 2 = ± 4
Solving for x Now, we solve for x by subtracting 2 from both sides: x = − 2 ± 4
Finding the Solutions Finally, we find the two possible values for x :
x = − 2 + 4 = 2 x = − 2 − 4 = − 6 So, the solutions are x = 2 and x = − 6 .
Final Answer The solutions to the equation x 2 + 4 x − 4 = 8 are x = 2 and x = − 6 .
Examples
Completing the square is a useful technique in various real-world scenarios. For instance, consider optimizing the area of a rectangular garden given a fixed perimeter. By expressing the area in terms of one side and completing the square, you can find the dimensions that maximize the garden's area. This method helps in resource allocation and optimization problems, ensuring the best possible outcome under given constraints.
