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Complete the square to identify the two values that solve the following quadratic equation: x^2+8x=9

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3 answers

To solve the quadratic equation \( x^2 + 8x = 9 \) by completing the square, follow these steps:

  1. Rearrange the equation so that the constant is on the right side: \[ x^2 + 8x - 9 = 0 \]

  2. Move the constant to the other side: \[ x^2 + 8x = 9 \]

  3. Complete the square on the left side. To do this, take half of the coefficient of \( x \) (which is 8), square it, and add it to both sides:

    • Half of 8 is 4, and \( 4^2 = 16 \). \[ x^2 + 8x + 16 = 9 + 16 \]

  4. Simplify both sides: \[ (x + 4)^2 = 25 \]

  5. Take the square root of both sides: \[ x + 4 = \pm 5 \]

  6. Solve for \( x \) by isolating it:

    • First solution: \[ x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1 \]
    • Second solution: \[ x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9 \]

Thus, the two values that solve the equation \( x^2 + 8x = 9 \) are: \[ \boxed{1} \quad \text{and} \quad \boxed{-9}. \]

there is not one answer?

The quadratic equation \(x^2 + 8x = 9\) actually has two solutions because it is a second-degree polynomial equation.

After completing the square, we found that: \[ (x + 4)^2 = 25 \] Taking the square root gives us two cases:

  1. \(x + 4 = 5\)
  2. \(x + 4 = -5\)

Solving these gives:

  1. From \(x + 4 = 5\): \[ x = 1 \]
  2. From \(x + 4 = -5\): \[ x = -9 \]

Thus, the two solutions are \(x = 1\) and \(x = -9\).

If you need just one answer, you can refer to either \(x = 1\) or \(x = -9\), but keep in mind that both values are solutions to the equation. If you're looking for a specific format or context, please let me know!