Solve each equation in Exercises 1 - 14 by factoring.6x^2 + 11x - 10 = 0

Verified step by step guidance

1. Step 1: Identify the quadratic equation in standard form. The given equation is already in the form \( ax^2 + bx + c = 0 \), where \( a = 6 \), \( b = 11 \), and \( c = -10 \).
2. Step 2: Look for two numbers that multiply to \( a \times c = 6 \times (-10) = -60 \) and add up to \( b = 11 \).
3. Step 3: The numbers that satisfy these conditions are 15 and -4, because \( 15 \times (-4) = -60 \) and \( 15 + (-4) = 11 \).
4. Step 4: Rewrite the middle term \( 11x \) using the numbers found: \( 6x^2 + 15x - 4x - 10 = 0 \).
5. Step 5: Factor by grouping. Group the terms: \((6x^2 + 15x) + (-4x - 10) = 0\). Factor out the greatest common factor from each group: \(3x(2x + 5) - 2(2x + 5) = 0\). Notice that \((2x + 5)\) is a common factor, so factor it out: \((3x - 2)(2x + 5) = 0\).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting the equation in the form of a product of two binomials. This process helps to find the roots of the equation by setting each factor equal to zero. For example, the equation ax^2 + bx + c can often be expressed as (px + q)(rx + s) = 0, where p, q, r, and s are constants.

Zero Product Property

The Zero Product Property states that if the product of two factors equals zero, then at least one of the factors must be zero. This principle is crucial when solving factored equations, as it allows us to set each factor to zero and solve for the variable. For instance, if (x - 2)(x + 3) = 0, then either x - 2 = 0 or x + 3 = 0.

Quadratic Formula

The Quadratic Formula provides a method for finding the roots of any quadratic equation in the standard form ax^2 + bx + c = 0. The formula is x = (-b ± √(b² - 4ac)) / (2a). While factoring is often preferred for simpler equations, the quadratic formula is a reliable alternative when factoring is difficult or impossible.