Question

Put the quadratic into vertex form and state the coordinates of the vertex.
y, equals, x, squared, plus, 10, x, plus, 30
y=
x
2
+10x+30

Answer 1

To convert the quadratic equation \( y = x^2 + 10x + 30 \) into vertex form, we will complete the square.

1. Start with the equation:
\[
y = x^2 + 10x + 30
\]

2. Focus on the quadratic and linear terms \( x^2 + 10x \). To complete the square, take half of the coefficient of \( x \) (which is \( 10 \)), square it, and add it:
- Half of \( 10 \) is \( 5 \), and \( 5^2 = 25 \).

3. Add and subtract \( 25 \) inside the equation:
\[
y = (x^2 + 10x + 25) - 25 + 30
\]

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

Now that we have the equation in vertex form \( y = a(x - h)^2 + k \), where \( (h, k) \) is the vertex:

- The vertex is at \( (-5, 5) \).

Therefore, the vertex form of the quadratic is:
\[
y = (x + 5)^2 + 5
\]
And the coordinates of the vertex are:
\[
\text{Vertex: } (-5, 5)
\]