To solve the equation \( y = a x^2 + c \) for \( x \), we need to isolate \( x \) step-by-step. Let's go through the process:
1. Start with the given equation:
[tex]\[ y = a x^2 + c \][/tex]
2. Isolate the \( x^2 \) term:
[tex]\[ y - c = a x^2 \][/tex]
3. Divide both sides by \( a \):
[tex]\[ \frac{y - c}{a} = x^2 \][/tex]
4. Take the square root of both sides to solve for \( x \):
Since taking the square root can result in both positive and negative roots, we get:
[tex]\[ x = \pm \sqrt{\frac{y - c}{a}} \][/tex]
So, the correctly solved equation for \( x \) is:
[tex]\[ x = \pm \sqrt{\frac{y - c}{a}} \][/tex]
Let's cross-check the provided options:
- \( x = \pm \sqrt{a y - c} \): This is not correct because it suggests multiplication inside the square root in a form not derived from the equation.
- \( x = \pm \sqrt{\frac{y - c}{a}} \): This is indeed the correct solution.
- \( x = \sqrt{\frac{y}{a} - c} \): This is wrong because the term \( c \) should be adjusted for multiplication/division prior to isolating \( x \).
- \( x = \sqrt{\frac{y + c}{a}} \): This is incorrect since subtracting \( c \) from \( y \) is the correct operation.
Therefore, the only correct expression for \( x \) is:
[tex]\[ x = \pm \sqrt{\frac{y - c}{a}} \][/tex]
