Solve $x^2+4 x+12=0$ by completing the square.
A. $x=-2+2 \sqrt{2} i, x=-2-2 \sqrt{2} i$
B. $x=-2+\sqrt{2} i, x=-2-\sqrt{2} i$
C. $x=-2+2 i, x=-2-2 i$
D. $x=0,-4$
Answer (1)
x = − 2 + 2 2 i , x = − 2 − 2 2 i
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 4 x + 12 = 0 and asked to solve it by completing the square. Completing the square is a method used to rewrite a quadratic equation in a form that allows us to easily find the solutions.
Completing the Square To complete the square, we want to rewrite the quadratic expression x 2 + 4 x + 12 in the form ( x + a ) 2 + b , where a and b are constants. We know that ( x + a ) 2 = x 2 + 2 a x + a 2 . Comparing this with x 2 + 4 x + 12 , we see that 2 a = 4 , so a = 2 . Thus, we want to rewrite the expression as ( x + 2 ) 2 + b .
Rewriting the Equation Expanding ( x + 2 ) 2 , we get x 2 + 4 x + 4 . Now we can rewrite the original equation as follows:
x 2 + 4 x + 12 = ( x 2 + 4 x + 4 ) + 8 = ( x + 2 ) 2 + 8
So, the equation becomes ( x + 2 ) 2 + 8 = 0 .
Isolating the Squared Term Now, we isolate the squared term:
( x + 2 ) 2 = − 8
Taking the square root of both sides, we get:
x + 2 = × \t s q r t − 8 = × \t s q r t 8 i = × 2 \t s q r t 2 i
Solving for x Finally, we solve for x :
x = − 2 × 2 \t s q r t 2 i
Thus, the solutions are x = − 2 + 2 \t s q r t 2 i and x = − 2 − 2 \t s q r t 2 i .
Final Answer Comparing our solution with the provided options, we see that the correct solutions are x = − 2 + 2 \t s q r t 2 i and x = − 2 − 2 \t s q r t 2 i .
Examples
Completing the square is a useful technique in various fields, such as physics and engineering, where quadratic equations often arise. For example, in projectile motion, the height of an object can be modeled by a quadratic equation, and completing the square can help determine the maximum height reached by the object. Similarly, in electrical circuits, quadratic equations can be used to analyze the behavior of circuits, and completing the square can help find the values of circuit parameters that optimize performance. This method provides a structured way to solve quadratic equations and gain insights into the underlying physical systems.
