Question

Complete the square to re-write the quadratic function in vertex form:
y, equals, x, squared, plus, 4, x, minus, 2
y=x
2
+4x−2

Answer 1

To rewrite the quadratic function \( y = x^2 + 4x - 2 \) in vertex form, we will complete the square.

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

2. **Isolate the constant term:**
\[
y = (x^2 + 4x) - 2
\]

3. **Complete the square for the expression \( x^2 + 4x \):**
- Take half of the coefficient of \( x \) (which is 4), square it, and add/subtract that inside the parentheses:
\[
\left(\frac{4}{2}\right)^2 = 2^2 = 4
\]

4. **Add and subtract this square inside the parentheses:**
\[
y = (x^2 + 4x + 4 - 4) - 2
\]
\[
y = ((x + 2)^2 - 4) - 2
\]

5. **Simplify the equation:**
\[
y = (x + 2)^2 - 4 - 2
\]
\[
y = (x + 2)^2 - 6
\]

Now the quadratic function in vertex form is:
\[
y = (x + 2)^2 - 6
\]

The vertex of the parabola represented by this function is at the point \((-2, -6)\).