Let's solve the equation [tex]\( y = ax^2 + c \)[/tex] for [tex]\( x \)[/tex] step by step.
1. Isolate the term involving [tex]\( x \)[/tex]:
[tex]\[
y = ax^2 + c
\][/tex]
Subtract [tex]\( c \)[/tex] from both sides:
[tex]\[
y - c = ax^2
\][/tex]
2. Solve for [tex]\( x^2 \)[/tex]:
Divide both sides by [tex]\( a \)[/tex]:
[tex]\[
\frac{y - c}{a} = x^2
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides. Remember, taking the square root yields two solutions, corresponding to the positive and negative square roots:
[tex]\[
x = \pm \sqrt{\frac{y - c}{a}}
\][/tex]
Therefore, the solutions for [tex]\( x \)[/tex] are:
[tex]\[
x = \sqrt{\frac{y - c}{a}} \quad \text{and} \quad x = -\sqrt{\frac{y - c}{a}}
\][/tex]
So, the complete solution to the equation [tex]\( y = ax^2 + c \)[/tex] for [tex]\( x \)[/tex] is:
[tex]\[
x = \pm \sqrt{\frac{y - c}{a}}
\][/tex]
