Solve each quadratic equation using the zero-factor property. See Example 5. 25x² + 30x + 9 = 0

Verified step by step guidance

1. Start with the quadratic equation: \$25x^{2} + 30x + 9 = 0$.
2. Check if the quadratic can be factored into the form \((ax + b)(cx + d) = 0\) by looking for two numbers that multiply to \(25 \times 9 = 225\) and add to \(30\).
3. Rewrite the middle term \$30x$ as a sum of two terms using the numbers found in the previous step, then group the terms to factor by grouping.
4. Factor out the greatest common factor (GCF) from each group to express the quadratic as a product of two binomials.
5. Apply the zero-factor property: set each binomial equal to zero, \(ax + b = 0\) and \(cx + d = 0\), then solve each equation for \(x\) to find the solutions.

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 the values of x that satisfy the equation.

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 is used to solve quadratic equations by factoring them into two binomials set equal to zero.

Factoring Quadratic Expressions

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