To convert the equation of a parabola from vertex form to standard form, we need to expand and simplify the expression.
Given vertex form:
[tex]\[ y = (x + 5)^2 + 49 \][/tex]
First, we will expand the [tex]\((x + 5)^2\)[/tex] term using the distributive property or the formula for squaring a binomial [tex]\((a + b)^2 = a^2 + 2ab + b^2\)[/tex].
[tex]\[ (x + 5)^2 = x^2 + 2 \cdot x \cdot 5 + 5^2 \][/tex]
[tex]\[ (x + 5)^2 = x^2 + 10x + 25 \][/tex]
Now, substitute this expanded form back into the original equation:
[tex]\[ y = (x + 5)^2 + 49 \][/tex]
[tex]\[ y = x^2 + 10x + 25 + 49 \][/tex]
Next, combine the constant terms (25 and 49):
[tex]\[ 25 + 49 = 74 \][/tex]
So the equation in standard form is:
[tex]\[ y = x^2 + 10x + 74 \][/tex]
Therefore, the standard form of the equation is:
[tex]\[ y = x^2 + 10x + 74 \][/tex]
The correct answer is:
C. [tex]\( y = x^2 + 10x + 74 \)[/tex]
