Solve each equation in Exercises 83–108 by the method of your choice. $2x^2-7x=0$

Verified step by step guidance

1. Start with the given quadratic equation: \$2x^2 - 7x = 0$.
2. Factor out the greatest common factor (GCF) from the terms on the left side. Here, \(x\) is common, so rewrite the equation as \(x(2x - 7) = 0\).
3. Apply the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero: \(x = 0\) and \$2x - 7 = 0$.
4. Solve each equation separately. The first is already solved: \(x = 0\). For the second, add 7 to both sides to get \$2x = 7\(, then divide both sides by 2 to isolate \)x\(: \)x = \frac{7}{2}$.
5. Write the solution set as \(\{0, \frac{7}{2}\}\), which includes all values of \(x\) that satisfy the original equation.

Key Concepts

Factoring Quadratic Equations

Factoring involves rewriting a quadratic equation as a product of simpler expressions set equal to zero. This method is useful when the quadratic can be expressed as a product of binomials or a monomial and a binomial, allowing the use of the zero-product property to find solutions.

Zero-Product Property

The zero-product property states that if the product of two factors is zero, then at least one of the factors must be zero. This principle is essential for solving equations after factoring, as it allows setting each factor equal to zero to find the roots of the equation.

Solving Linear Equations

After factoring, some factors may be linear expressions. Solving linear equations involves isolating the variable on one side to find its value. This step is straightforward and necessary to determine all solutions of the original quadratic equation.