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Solve [tex]$y = a x^2 + c[tex]$[/tex] for [tex]$[/tex]x$[/tex].

A. [tex]$x = \pm \sqrt{a y - c}$[/tex]
B. [tex]$x = \pm \sqrt{\frac{y - c}{a}}$[/tex]
C. [tex]$x = \sqrt{\frac{y}{a} - c}$[/tex]
D. [tex]$x = \sqrt{\frac{y + c}{a}}$[/tex]

Sagot :

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]
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