Question

Find the zeros of y = x^2 4x - 9 by completing the square

Answer 1

To find the zeros of the quadratic equation \( y = x^2 + 4x - 9 \) by completing the square, we follow these steps:

1. **Start with the quadratic equation:**
\[
y = x^2 + 4x - 9.
\]

2. **Rearrange the equation:**
We want to focus on the terms involving \( x \). First, we can isolate the constant on the right side:
\[
x^2 + 4x = y + 9.
\]

3. **Complete the square:**
To complete the square for the left side \( x^2 + 4x \):
- Take half of the coefficient of \( x \) (which is 4), square it:
\[
\left(\frac{4}{2}\right)^2 = 4.
\]
- Add and subtract this square inside the equation:
\[
x^2 + 4x + 4 - 4 = y + 9.
\]
- This can be rearranged as:
\[
(x + 2)^2 - 4 = y + 9.
\]

4. **Simplify the equation:**
Now we can write:
\[
(x + 2)^2 - 4 = y + 9,
\]
which gives us
\[
(x + 2)^2 = y + 13.
\]

5. **Set \( y \) to zero to find the zeros:**
To find the zeros, set \( y = 0 \):
\[
(x + 2)^2 = 13.
\]

6. **Solve for \( x \):**
Take the square root of both sides:
\[
x + 2 = \pm \sqrt{13}.
\]
Then we solve for \( x \):
\[
x = -2 + \sqrt{13} \quad \text{or} \quad x = -2 - \sqrt{13}.
\]

Thus, the zeros of the equation \( y = x^2 + 4x - 9 \) are:
\[
x = -2 + \sqrt{13} \quad \text{and} \quad x = -2 - \sqrt{13}.
\]