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

Verified step by step guidance

1. Combine like terms on the left side of the equation: \( a^2x + 3x = 2a^2 \).
2. Factor out \( x \) from the left side: \( x(a^2 + 3) = 2a^2 \).
3. Isolate \( x \) by dividing both sides by \( a^2 + 3 \): \( x = \frac{2a^2}{a^2 + 3} \).
4. Ensure that \( a^2 + 3 \neq 0 \) to avoid division by zero.
5. The solution for \( x \) is expressed in terms of \( a \) as \( x = \frac{2a^2}{a^2 + 3} \).

Key Concepts

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together give the original expression. In the context of equations, factoring can help isolate the variable, making it easier to solve for its value. For example, in the equation a²x + 3x = 2a², factoring out the common term x can simplify the equation significantly.

Linear Equations

A linear equation is an equation of the first degree, meaning it involves only the first power of the variable. These equations can typically be expressed in the form ax + b = 0, where a and b are constants. Understanding linear equations is crucial for solving for x, as they represent straight lines on a graph and have a single solution unless they are parallel or identical.

Isolating the Variable

Isolating the variable is a fundamental technique in algebra where the goal is to get the variable (in this case, x) on one side of the equation by itself. This often involves performing inverse operations, such as addition, subtraction, multiplication, or division, to both sides of the equation. Successfully isolating the variable allows for straightforward calculation of its value.