To solve the equation [tex]\( y = a x^2 + c \)[/tex] for [tex]\( x \)[/tex], let's follow a step-by-step process:
1. Isolate the quadratic term:
Begin by isolating the term with [tex]\( x^2 \)[/tex].
[tex]\[
y = a x^2 + c
\][/tex]
Subtract [tex]\( c \)[/tex] from both sides to isolate [tex]\( a x^2 \)[/tex]:
[tex]\[
y - c = a x^2
\][/tex]
2. Solve for [tex]\( x^2 \)[/tex]:
Divide both sides by [tex]\( a \)[/tex] to solve for [tex]\( x^2 \)[/tex]:
[tex]\[
x^2 = \frac{y - c}{a}
\][/tex]
3. Simplify the expression for [tex]\( x \)[/tex]:
Take the square root of both sides to solve for [tex]\( x \)[/tex]. Remember that taking the square root of both sides will give both the positive and negative roots.
[tex]\[
x = \pm \sqrt{\frac{y - c}{a}}
\][/tex]
Let's review the given multiple choice options to find a match to our derived solution:
1. [tex]\( x = \pm \sqrt{a y - c} \)[/tex]
This option is incorrect because it does not correctly isolate and simplify the equation for [tex]\( x \)[/tex].
2. [tex]\( x = \pm \sqrt{\frac{y - c}{a}} \)[/tex]
This option matches our derived solution exactly, thus it is correct.
3. [tex]\( x = \sqrt{\frac{y}{a} - c} \)[/tex]
This option is incorrect because it incorrectly handles the separation of terms in the denominator.
4. [tex]\( x = \sqrt{\frac{y + c}{a}} \)[/tex]
This option is incorrect because it incorrectly adds [tex]\( c \)[/tex] inside the square root instead of subtracting it before dividing by [tex]\( a \)[/tex].
Therefore, the correct solution based on our careful analysis and step-by-step process is:
[tex]\[ x = \pm \sqrt{\frac{y - c}{a}} \][/tex]
Hence, the correct answer is:
[tex]\[ 2 \][/tex]
