Answer:
The vertex is (3,4)
Step-by-step explanation:
To convert a quadratic from
y
=
a
x
2
+
b
x
+
c
form to vertex form,
y
=
a
(
x
−
h
)
2
+
k
, you use the process of completing the square.
First, we must isolate the x
terms:
y
−
49
=
5
x
2
−
30
x
+
49
−
49
y
−
49
=
5
x
2
−
30
x
We need a leading coefficient of 1
for completing the square, so factor out the current leading coefficient of 2.
y
−
49
=
5
(
x
2
−
6
x
)
Next, we need to add the correct number to both sides of the equation to create a perfect square. However, because the number will be placed inside the parenthesis on the right side we must factor it by
2
on the left side of the equation. This is the coefficient we factored out in the previous step.
y
−
49
+
(
5
⋅
?
)
=
5
(
x
2
−
6
x
+
?
)
<- Hint:
62
=
3
; 3
⋅
3
=
9
y
−
49
+
(
5
⋅
9
)
=
5
(
x
2
−
6
x
+
9
)
y−
49
+
45
=
5
(
x
2
−
6
x
+
9
)
y
−
4
=
5
(
x
2
−
6
x
+
9
)
Then, we need to create the square on the right hand side of the equation:
y
−
4
=
5
(
x
−
3
)
2
Now, isolate the y term:
y
−
4
+
4
=
5
(
x
−
3
)
2
+
4
y
−
0
=
5
(
x
−
3
)
2
+
4
y
−
0
=
5
(
x
−
3
)
2
+
4
The vertex is:
(
3
,
4
)
