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

Verified step by step guidance

Start with the given equation: $a^{2}x + 3x = 2a^{2}$.

Factor out the common factor $x$ on the left side: $x(a^{2} + 3) = 2a^{2}$.

To isolate $x$, divide both sides of the equation by the quantity $(a^{2} + 3)$: $x = \frac{2a^{2}}{a^{2} + 3}$.

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.

Thus, the solution for $x$ is expressed as $x = \frac{2a^{2}}{a^{2} + 3}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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