Complete the square to identify the two values that solve the following quadratic equation: x2+8x=9 .

Answer 1

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

1. Rearrange the equation: Move the constant term to the left side.

   \[ x^2 + 8x - 9 = 0 \]

2. Isolate the constant term: We want the left side to represent a perfect square.

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

3. Complete the square: Take the coefficient of \( x \) (which is 8), divide it by 2 (giving us 4), and square it (resulting in 16). Now, add and subtract this square inside the equation.

   \[ x^2 + 8x + 16 - 16 = 9 \]

   This can be rewritten as:

   \[ (x + 4)^2 - 16 = 9 \]

4. Rearrange it to a standard form:

   \[ (x + 4)^2 = 9 + 16 \]

   \[ (x + 4)^2 = 25 \]

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

   \[ x + 4 = 5 \quad \text{or} \quad x + 4 = -5 \]

6. Solve for \( x \):

   • For \( x + 4 = 5 \):

     \[ x = 5 - 4 = 1 \]

   • For \( x + 4 = -5 \):

     \[ x = -5 - 4 = -9 \]

So, the two solutions to the equation \( x^2 + 8x = 9 \) are:

\[ \boxed{1} \quad \text{and} \quad \boxed{-9} \]