Solve each equation for x. a²x + 3x =2a²

Verified step by step guidance

1. Start with the given equation: \(a^{2}x + 3x = 2a^{2}\).
2. Factor out the common factor \(x\) on the left side: \(x(a^{2} + 3) = 2a^{2}\).
3. To isolate \(x\), divide both sides of the equation by the quantity \((a^{2} + 3)\): \(x = \frac{2a^{2}}{a^{2} + 3}\).
4. Check the denominator to ensure it is not zero, which would make the expression undefined. Since \(a^{2} + 3\) is always positive for all real \(a\), division is valid.
5. Thus, the solution for \(x\) is expressed as \(x = \frac{2a^{2}}{a^{2} + 3}\).

Key Concepts

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. In the equation a²x + 3x = 2a², both terms on the left contain x, so they can be combined by factoring x out, simplifying the expression for easier solving.

Factoring

Factoring is the process of expressing an expression as a product of its factors. Here, factoring x from a²x + 3x gives x(a² + 3), which simplifies the equation and allows isolating x by dividing both sides by (a² + 3), assuming it is not zero.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side to find its value. After factoring, the equation becomes x(a² + 3) = 2a², and dividing both sides by (a² + 3) yields x = 2a² / (a² + 3), provided the denominator is not zero.