Solve each equation in Exercises 1 - 14 by factoring. $2 x (x - 3) = 5 x^{2} - 7 x$

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Start by expanding the left side of the equation: multiply 2x by each term inside the parentheses to get $2x \cdot x$ and $2x \cdot (-3)$, which gives $2x^{2} - 6x$.

Rewrite the equation with the expanded left side: $2x^{2} - 6x = 5x^{2} - 7x$.

Bring all terms to one side to set the equation equal to zero. Subtract \$5x^{2}$ and add $7x$ to both sides: $2x^{2} - 6x - 5x^{2} + 7x = 0$.

Combine like terms: $2x^{2} - 5x^{2} = -3x^{2}$ and $-6x + 7x = x$, so the equation becomes $-3x^{2} + x = 0$.

Factor the resulting expression by taking out the greatest common factor (GCF), which is $x$: $x(-3x + 1) = 0$. Then, set each factor equal to zero to find the solutions.

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Key Concepts

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

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of simpler binomials or monomials. This process helps in solving equations by setting each factor equal to zero, making it easier to find the roots of the equation.

Setting the Equation to Zero

To solve an equation by factoring, first rearrange all terms so that one side equals zero. This standard form allows the use of the zero-product property, which states that if a product of factors is zero, at least one factor must be zero.

Zero-Product Property

The zero-product property states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle is essential for solving factored equations by setting each factor equal to zero and solving for the variable.

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