The equation for \(y = 3x^2 + 5\) represents a quadratic function. Which of the following equations isolates the variable \(x\)?
Understand the Problem
The question provides a quadratic equation, (y = 3x^2 + 5), and asks to identify the equation that correctly isolates (x). We need to rearrange the equation to solve for x, by subtracting 5 and dividing by 3, and then taking the square root.
Answer
$x = \pm \sqrt{\frac{y - 5}{3}}$
Answer for screen readers
$x = \pm \sqrt{\frac{y - 5}{3}}$
Steps to Solve
- Isolate the term with $x$
Start with the given equation: $y = 3x^2 + 5$ Subtract 5 from both sides: $y - 5 = 3x^2$
- Isolate $x^2$
Divide both sides by 3: $\frac{y - 5}{3} = x^2$
- Solve for $x$
Take the square root of both sides: $x = \pm \sqrt{\frac{y - 5}{3}}$
$x = \pm \sqrt{\frac{y - 5}{3}}$
More Information
When taking the square root to solve for $x$, remember both positive and negative roots are possible.
Tips
A common mistake is forgetting to include both the positive and negative square roots when solving for $x$. Another common mistake is incorrectly isolating terms, such as subtracting 5 after dividing by 3, or not performing the operations in the correct order.
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