Solving the equation [tex]\( y = a x^2 + c \)[/tex] for [tex]\( x \)[/tex]:
1. Start with the given equation:
[tex]\[
y = a x^2 + c
\][/tex]
2. Isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y - c = a x^2
\][/tex]
3. Solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{y - c}{a}
\][/tex]
4. Take the square root of both sides to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \pm \sqrt{\frac{y - c}{a}}
\][/tex]
So, the solution for [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex], [tex]\( a \)[/tex], and [tex]\( c \)[/tex] is:
[tex]\[
x = \pm \sqrt{\frac{y - c}{a}}
\][/tex]
After comparing with the given options:
1. [tex]\( x = \pm \sqrt{a y - c} \)[/tex]
2. [tex]\( x = \pm \sqrt{\frac{y - c}{a}} \)[/tex]
3. [tex]\( x = \sqrt{\frac{y}{a} - c} \)[/tex]
4. [tex]\( x = \sqrt{\frac{y + c}{a}} \)[/tex]
The correct form aligns with option 2.
Hence, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
