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Factoring algebraic expressions involves rewriting a polynomial as a product of simpler polynomials. This method can help in solving quadratic equations. Let's see how factoring works in the given example.
Initially, the equation is: \(3x^2 = 12x\). We need to get all terms on one side, so we subtract 12x from both sides:
\[3x^2 - 12x = 0\]
Next, we look for the greatest common factor (GCF) in the terms. Here, both terms share a factor of 3x.
When we factor out 3x, we get: \[3x(x - 4) = 0\]
Factoring simplifies the expression, making it easier to apply the zero product property. Always check each term for common factors and factor them out wherever possible.

When solving quadratic equations, factoring is one of the most useful methods. A quadratic equation has the form \(ax^2 + bx + c = 0\). To solve it by factoring, we first move all terms to one side of the equation to set it equal to zero.
Then, we factor the quadratic expression. In the zero product property, if a product of multiple terms equals zero, then at least one of the terms must be zero. For the equation \[3x(x - 4) = 0\] we use this property.
We set each factor to zero:
\[3x = 0 \quad \text{or} \quad x-4 = 0\]
Solving these, we find:
\[3x = 0 \implies x = 0\] and
\[x-4 = 0 \implies x = 4\]
So, the solutions to the quadratic equation are \(x = 0\) and \(x = 4\). This approach is very systematic and helps students solve a range of quadratic equations.

The greatest common factor (GCF) is the largest number or expression that can divide two or more terms without leaving a remainder. Finding the GCF is crucial when factoring algebraic expressions.
In our problem, we have: \(3x^2 - 12x\).
To factor this expression, spot common factors in the terms. Here, both 3 and 12 are divisible by 3, and both terms contain the variable x. Thus, the GCF is 3x.
Factoring out the GCF, we get: \[3x(x - 4) = 0\]
Without factoring the GCF, it would be impossible to apply the zero product property effectively. Always identify and factor the GCF first to simplify the equation and make solving easier.