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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

Question

Solve the equation for all values of x by completing the square.

Given the equation:
x2+14x+48=0 x^2 + 14x + 48 = 0

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Solution

1. Break Down the Problem

We need to solve the quadratic equation x2+14x+48=0 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 (x + p)^2 = q .

3. Analysis and Detail

Step 1: Rearrange the equation x2+14x+48=0 x^2 + 14x + 48 = 0 To complete the square, we need to isolate the constant term: x2+14x=48 x^2 + 14x = -48

Step 2: Find the term to complete the square Take half of the coefficient of x x (which is 14), square it, and add to both sides: (142)2=72=49 \left( \frac{14}{2} \right)^2 = 7^2 = 49 Add 49 to both sides: x2+14x+49=48+49 x^2 + 14x + 49 = -48 + 49 This simplifies to: x2+14x+49=1 x^2 + 14x + 49 = 1

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

4. Verify and Summarize

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

Step 5: Solve for x x

  1. x+7=1 x + 7 = 1 leads to: x=17=6 x = 1 - 7 = -6
  2. x+7=1 x + 7 = -1 leads to: x=17=8 x = -1 - 7 = -8

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

Final Answer

The solutions for the equation x2+14x+48=0 x^2 + 14x + 48 = 0 are: x=6andx=8 x = -6 \quad \text{and} \quad x = -8

This problem has been solved

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