How can we solve the equation x² + 6x + 7 by completing the square?

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To solve the quadratic equation x² + 6x + 7 = 0 by completing the square, we’ll follow these steps:

  1. Rearrange the equation: Start by moving the constant term to the right side of the equation:

    2. x² + 6x = -7

  2. Complete the square: To complete the square, we need to turn the left side of the equation into a perfect square trinomial. The coefficient of x is 6, and half of 6 is 3. Now, square this value:

    3. (3)² = 9

    Next, add this square to both sides of the equation:

    x² + 6x + 9 = -7 + 9

    This simplifies to:

    (x + 3)² = 2

  1. Take the square root of both sides: To solve for x, take the square root of both sides. Remember to consider both the positive and negative square roots:

    2. x + 3 = ±√2

  1. Isolate x: Finally, solve for x by isolating it:

    2. x = -3 ± √2

Thus, the solution set of the equation x² + 6x + 7 = 0 is:

{ -3 + √2, -3 – √2 }

In conclusion, by completing the square, we’ve transformed the quadratic into a more manageable format and solved for x, revealing our solution set!

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