Question

vertex form of y=x^2-2x-5

Answer 1

To convert the quadratic equation \( y = x^2 - 2x - 5 \) into vertex form, we can complete the square. The vertex form of a quadratic equation is given by:

\[
y = a(x-h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

### Step 1: Factor out the coefficient of \(x^2\) (which is 1 in this case)

Since the coefficient of \(x^2\) is 1, we can proceed without factoring.

### Step 2: Complete the square

1. Take the coefficient of \(x\) (which is \(-2\)), divide it by 2, and square it:
\[
\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1
\]

2. Add and subtract this square inside the equation:
\[
y = x^2 - 2x + 1 - 1 - 5
\]

This simplifies to:
\[
y = (x - 1)^2 - 6
\]

### Step 3: Write in vertex form

Now we have the equation in vertex form:
\[
y = (x - 1)^2 - 6
\]

So, the vertex of the parabola is \((h, k) = (1, -6)\).