For the following exercises, solve the equation by...

For the following exercises, solve the equation by factoring. You will need to set the equation equal to 0 first, then find the greatest common factor, and finally see if further factoring is possible.
$3 x^2=75$

Key Concepts

Quadratic Equations

Quadratic equations are second-degree polynomial equations that can often be solved by factoring, among other methods such as completing the square or using the quadratic formula. Understanding the structure of quadratic equations and how to manipulate them, including setting them to zero and factoring, is a key algebraic skill.

Factoring

Factoring is the process of breaking down an expression into a product of simpler expressions or factors. It is an essential technique for solving quadratic equations and other higher-order equations. The method of factoring facilitates the use of the zero product property, allowing each factor to be set to zero to solve for the variable.

Greatest Common Factor (GCF)

The greatest common factor is the largest expression that divides all the terms in a polynomial without a remainder. Identifying and factoring out the GCF simplifies the polynomial, making further factoring easier. This step reduces the complexity of the polynomial and can reveal hidden patterns that enable complete factorization.

Equation Balancing and Zeroing

This concept involves rewriting an equation such that one side becomes zero. It is a fundamental step in solving algebraic equations, especially when using factoring techniques. By rearranging the equation to have zero on one side, it becomes possible to apply the zero product property, which is critical in finding the solutions.

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