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Solve x2 + 12x = –20 by completing the square.

Add (StartFraction b Over 2 EndFraction) squared to both sides of the equation. The value of (StartFraction b Over 2 EndFraction) squared in this equation is
.
Write the left side of the equation as a binomial squared. The left side of the equation becomes (
)2.
Use the square root property of equality.
Isolate the variable: x =

1 answer

To solve the equation \( x^2 + 12x = -20 \) by completing the square, follow these steps:

  1. Move the constant term to the other side: \[ x^2 + 12x + 20 = 0 \] (We can actually just keep the equation \( x^2 + 12x = -20 \) and add 20 to both sides later, but moving it here clarifies the next step.)

  2. Identify \( b \): In the equation \( x^2 + 12x \), \( b = 12 \).

  3. Calculate \( \left( \frac{b}{2} \right)^2 \): \[ \left( \frac{12}{2} \right)^2 = 6^2 = 36 \]

  4. Add \( \left( \frac{b}{2} \right)^2 \) to both sides of the equation: \[ x^2 + 12x + 36 = -20 + 36 \] \[ x^2 + 12x + 36 = 16 \]

  5. Write the left side as a binomial squared: \[ (x + 6)^2 = 16 \]

  6. Use the square root property of equality: \[ x + 6 = \pm 4 \]

  7. Isolate the variable:

    • For \( x + 6 = 4 \): \[ x = 4 - 6 = -2 \]
    • For \( x + 6 = -4 \): \[ x = -4 - 6 = -10 \]

Thus, the solutions to the equation \( x^2 + 12x = -20 \) are \( x = -2 \) and \( x = -10 \).