Solve each quadratic equation using the zero-factor property. See Example 5. 5x² - 3x - 2 = 0

Verified step by step guidance

1. Start with the quadratic equation: \$5x^{2} - 3x - 2 = 0$.
2. Factor the quadratic expression on the left side. Look for two numbers that multiply to \(5 \times (-2) = -10\) and add to \(-3\).
3. Rewrite the middle term \(-3x\) using the two numbers found, then group terms to factor by grouping.
4. Apply the zero-factor property, which states that if \(ab = 0\), then either \(a = 0\) or \(b = 0\). Set each factor equal to zero.
5. Solve each resulting linear equation for \(x\) to find the solutions to the quadratic equation.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of degree two, generally written as ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. Solving these equations involves finding values of x that satisfy the equation.

Factoring Quadratic Expressions

Factoring involves rewriting a quadratic expression as a product of two binomials. This step is essential to apply the zero-factor property, as it breaks down the quadratic into simpler expressions that can be set to zero.

Zero-Factor Property

The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property allows us to set each factor equal to zero and solve for the variable to find the roots of the equation.