Question

Transform the quadratic equation from standard form to vertex form.

y=x2+6x+5

(1 point)
The vertex form of the quadratic function is y=(x−
)2
+
.

Answer 1

To convert the quadratic equation from standard form \( y = x^2 + 6x + 5 \) to vertex form \( y = a(x - h)^2 + k \), we will complete the square.

1. Start with the original equation:
\[
y = x^2 + 6x + 5
\]

2. Focus on the quadratic and linear terms \( x^2 + 6x \). To complete the square, take half of the coefficient of \( x \) (which is 6), square it, and add and subtract this value inside the equation.
- Half of 6 is 3, and squaring it gives \( 3^2 = 9 \).

3. Rewrite the quadratic expression:
\[
y = (x^2 + 6x + 9) - 9 + 5
\]

4. Simplify the equation:
\[
y = (x + 3)^2 - 4
\]

Now, the equation is in vertex form.

Thus, the vertex form of the quadratic function is:
\[
y = (x + 3)^2 - 4
\]

From this, we can identify \( h = -3 \) and \( k = -4 \).

Therefore, the final answer is:
\[
\text{The vertex form of the quadratic function is } y = (x - (-3))^2 - 4.
\]

Which can also be expressed as:
\[
\text{The vertex form is } y = (x - (-3))^2 + (-4).
\]

Fill in the blanks accordingly:
- The vertex form of the quadratic function is \( y = (x - (-3))^2 - 4 \).