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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 complete the square for the equation \( x^2 + 8x = 9 \), we start by rearranging the equation:

  1. Move the constant to the other side: \[ x^2 + 8x - 9 = 0 \]

  2. To complete the square, we need to take the coefficient of \( x \) (which is 8), divide it by 2 (getting 4), and then square that value: \[ \left( \frac{8}{2} \right)^2 = 4^2 = 16 \]

  3. Now, add and subtract this value on the left side of the equation: \[ x^2 + 8x + 16 - 16 - 9 = 0 \]

  4. Simplify the left side: \[ (x + 4)^2 - 25 = 0 \]

  5. Move the constant term to the right side: \[ (x + 4)^2 = 25 \]

  6. Take the square root of both sides, remembering to consider both the positive and negative roots: \[ x + 4 = 5 \quad \text{or} \quad x + 4 = -5 \]

  7. Solving these two equations gives us: \[ x + 4 = 5 \quad \Rightarrow \quad x = 5 - 4 = 1 \] \[ x + 4 = -5 \quad \Rightarrow \quad x = -5 - 4 = -9 \]

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

it is a one answer thing

I understand you're looking for a singular solution based on the completed square method. Let's summarize the solution clearly in one concise form:

Starting with the equation:

\[ x^2 + 8x = 9 \]

We complete the square:

  1. Move 9 to the left: \[ x^2 + 8x - 9 = 0 \]

  2. Complete the square:

    • Take half of 8 (which is 4) and square it (which is 16).
    • Rewrite: \[ (x + 4)^2 - 16 - 9 = 0 \] \[ (x + 4)^2 - 25 = 0 \] \[ (x + 4)^2 = 25 \]

  3. Take the square root: \[ x + 4 = 5 \quad \text{or} \quad x + 4 = -5 \]

  4. Solve for \( x \):

    • From \( x + 4 = 5 \): \[ x = 1 \]

    • From \( x + 4 = -5 \): \[ x = -9 \]

Given you're looking for one answer, the solutions to the equation are \( x = 1 \) and \( x = -9 \).

If you're looking for a singular representation of the basic solution form, here it is:

The solutions are: \({1, -9}\).