How do you solve the quadratic equation x² + 10x + 24 by completing the square?

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To solve the quadratic equation x2 + 10x + 24 = 0 by completing the square, follow these steps:

  1. Move the constant to the other side: Start by subtracting 24 from both sides of the equation:

    2. x2 + 10x = -24

  2. Complete the square: To do this, take the coefficient of x (which is 10), divide it by 2 (resulting in 5), and then square it (resulting in 25). Add this square to both sides of the equation:

    3. x2 + 10x + 25 = -24 + 25

  3. Simplify both sides: This simplifies to:

    4. (x + 5)2 = 1

  4. Take the square root of both sides: Remember to consider both the positive and negative roots:

    5. x + 5 = ±1

  5. Isolate x: Solve for x in both cases:
    • Case 1: x + 5 = 1

      • This gives: x = 1 – 5 = -4

    • Case 2: x + 5 = -1

      • This gives: x = -1 – 5 = -6

  6. Conclusion: The solutions to the equation are:

    7. x = -4 and x = -6

Completing the square not only helps solve the equation, but it also provides a clear representation of the parabola’s vertex. In this case, the vertex lies at (-5, -1), adding a visual understanding of the quadratic function’s behavior!

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