Question

Transform the quadratic equation to vertex form and state the vertex point.

y=−5x^2+30x−35

(1 point)
Responses

y=5(x−3)^2+10
and (3,10)
y is equal to 5 times open paren x minus 3 close paren squared plus 10 and open paren 3 comma 10 close paren

y=−5(x−3)^2+10
and (−3,10)
y is equal to negative 5 times open paren x minus 3 close paren squared plus 10 and open paren negative 3 comma 10 close paren

y=−5(x−3)^2+10
and (3,10)
y is equal to negative 5 times open paren x minus 3 close paren squared plus 10 and open paren 3 comma 10 close paren

y=−5(x−3)^2−10
and (3,−10)

Answer 1

To convert the quadratic equation \( y = -5x^2 + 30x - 35 \) into vertex form, we can use the method of completing the square.

1. Start with the original equation:
\[
y = -5x^2 + 30x - 35
\]

2. Factor out \(-5\) from the terms involving \(x\):
\[
y = -5(x^2 - 6x) - 35
\]

3. To complete the square, take half of the coefficient of \(x\) (which is \(-6\)), square it, and add/subtract it within the parentheses:
- Half of \(-6\) is \(-3\), and \((-3)^2 = 9\).

Add and subtract this inside the parentheses:
\[
y = -5(x^2 - 6x + 9 - 9) - 35
\]
\[
y = -5((x - 3)^2 - 9) - 35
\]

4. Distribute \(-5\):
\[
y = -5(x - 3)^2 + 45 - 35
\]
\[
y = -5(x - 3)^2 + 10
\]

Now we have the vertex form:
\[
y = -5(x - 3)^2 + 10
\]

From this, we can see the vertex point is \((3, 10)\).

Thus, the correct response is:
**y = -5(x − 3)^2 + 10 and (3, 10)**.