To convert the equation of a parabola from standard form to vertex form, we start with the given standard form of the parabola:
[tex]\[ x = y^2 + 10y + 22 \][/tex]
Our goal is to complete the square on the \( y \)-terms to rewrite this equation in vertex form. The vertex form of a parabola is given by:
[tex]\[ x = a(y - h)^2 + k \][/tex]
where \((h, k)\) is the vertex of the parabola.
Here are the steps to transform the equation:
1. Isolate the quadratic and linear terms involving \( y \):
[tex]\[ x = y^2 + 10y + 22 \][/tex]
2. Set up for completing the square:
To complete the square, we take the coefficient of the linear term (10), halve it, and square the result. This gives:
[tex]\[ \left( \frac{10}{2} \right)^2 = 25 \][/tex]
3. Add and subtract this square inside the equation:
[tex]\[ x = y^2 + 10y + 25 - 25 + 22 \][/tex]
4. Reorganize and simplify:
Group the perfect square trinomial and simplify the constants:
[tex]\[ x = (y^2 + 10y + 25) - 25 + 22 \][/tex]
[tex]\[ x = (y + 5)^2 - 3 \][/tex]
We have successfully rewritten the equation in vertex form. Therefore, the vertex form of the equation is:
[tex]\[ x = (y + 5)^2 - 3 \][/tex]
Now, let's match this to the given options:
A. \( x = (y + 10)^2 - 78 \)
B. \( x = (y + 5)^2 - 3 \)
C. \( x = (y + 5)^2 + 12 \)
D. \( x = (y + 10)^2 + 2 \)
The correct answer is:
[tex]\[ \boxed{B} \][/tex]
Thus, the vertex form of the equation is [tex]\( x = (y + 5)^2 - 3 \)[/tex].
