Solve the equation for all values of x by completing the square.x, squared, plus, 14, x, plus, 48, equals, 0x 2 +14x+48=0

1. Break Down the Problem

We need to solve the quadratic equation $x^2 + 14x + 48 = 0$ by completing the square.

2. Relevant Concepts

The method of completing the square involves rearranging the quadratic equation into the form $(x + p)^2 = q$.

3. Analysis and Detail

Step 1: Rearrange the equation $x^2 + 14x + 48 = 0$ To complete the square, we need to isolate the constant term: $x^2 + 14x = -48$

Step 2: Find the term to complete the square Take half of the coefficient of $x$ (which is 14), square it, and add to both sides: $\left( \frac{14}{2} \right)^2 = 7^2 = 49$ Add 49 to both sides: $x^2 + 14x + 49 = -48 + 49$ This simplifies to: $x^2 + 14x + 49 = 1$

Step 3: Rewrite the left side as a squared term $(x + 7)^2 = 1$

4. Verify and Summarize

Step 4: Take the square root of both sides $x + 7 = \pm 1$

Step 5: Solve for $x$

1. $x + 7 = 1$ leads to: $x = 1 - 7 = -6$
2. $x + 7 = -1$ leads to: $x = -1 - 7 = -8$

The solutions to the equation are $x = -6$ and $x = -8$.

Final Answer

The solutions for the equation $x^2 + 14x + 48 = 0$ are: $x = -6 \quad \text{and} \quad x = -8$