91Ó°ÊÓ

Factoring is a critical step in solving quadratic equations. It involves breaking down a complex expression into simpler multiples that can be handled more easily. In the given problem, we started by identifying \(x + 3\) as a common factor. You can find the common factor by visually inspecting the terms on both sides of the equation.
If you see the same binomial (or sometimes a polynomial) multiplied by different coefficients, you have your common factor. Here, identifying \(x + 3\) helps in significantly simplifying our equation.
Factoring is powerful because it reduces the complexity of the equation. It transforms a quadratic equation into a simpler linear form, making it easier to solve.

Isolating the variable you are solving for is the next logical step after factoring. In this context, isolating the variable means we want to get \(x\) by itself on one side of the equation.
In our problem, we moved from \(2x(x+3) = (3x+1)(x+3)\) to \(2x = 3x + 1\) by dividing both sides by the common factor \(x + 3\). Once divided, we are left with a simpler equation. Try to always move terms involving the variable to one side.
Here’s what we did: \(2x - 3x = 1\). This basically means you have to subtract 3x from both sides to isolate the variable. Simple arithmetic then gives \( -x = 1\), and dividing both sides by \ -1 \ isolates \ x \ completely, giving us \ x = -1 \.

Verification of solutions is an important final step. This is to ensure that no mistakes have been made along the way.
To verify, substitute the solution back into the original equation. For our problem, we substituted \ x = -1 \ back: \(2(-1)(-1 + 3) = (3(-1) + 1)(-1 + 3) \), which simplifies to \(2(-1)(2) = (-3 + 1)(2) \). Both sides simplify to \ -4 = -4 \.
If both sides of the equation are equal, then the solution \ x = -1 \ is verified as correct. If not, it indicates that there might have been an error in the initial steps or in the arithmetic.
Always ensure the solution satisfies the original equation to confirm its validity.