Question
Convert the quadratic y = 2(x−5)2 − 10
from vertex form to standard form.(2 points)
The standard form of the quadratic is y=
x2+
x+
.
from vertex form to standard form.(2 points)
The standard form of the quadratic is y=
x2+
x+
.
Answers
GPT-4o mini
To convert the quadratic from vertex form to standard form, we start with the given vertex form of the quadratic:
\[ y = 2(x - 5)^2 - 10 \]
Now, we will expand and simplify this expression step by step.
1. First, expand the squared term:
\[
(x - 5)^2 = x^2 - 10x + 25
\]
2. Substitute this back into the equation:
\[
y = 2(x^2 - 10x + 25) - 10
\]
3. Distribute the 2:
\[
y = 2x^2 - 20x + 50 - 10
\]
4. Combine like terms:
\[
y = 2x^2 - 20x + 40
\]
Now, we have the quadratic in standard form \( y = ax^2 + bx + c \), where \( a = 2 \), \( b = -20 \), and \( c = 40 \).
Thus, the standard form of the quadratic is:
\[
y = 2x^2 - 20x + 40
\]
\[ y = 2(x - 5)^2 - 10 \]
Now, we will expand and simplify this expression step by step.
1. First, expand the squared term:
\[
(x - 5)^2 = x^2 - 10x + 25
\]
2. Substitute this back into the equation:
\[
y = 2(x^2 - 10x + 25) - 10
\]
3. Distribute the 2:
\[
y = 2x^2 - 20x + 50 - 10
\]
4. Combine like terms:
\[
y = 2x^2 - 20x + 40
\]
Now, we have the quadratic in standard form \( y = ax^2 + bx + c \), where \( a = 2 \), \( b = -20 \), and \( c = 40 \).
Thus, the standard form of the quadratic is:
\[
y = 2x^2 - 20x + 40
\]