Solve each equation in Exercises 1 - 14 by factoring. $x^2-3x-10=0$

Verified step by step guidance

1. Start with the quadratic equation: \(x^2 - 3x - 10 = 0\).
2. Look for two numbers that multiply to the constant term \(-10\) and add up to the coefficient of the linear term \(-3\).
3. Express the quadratic as a product of two binomials using those two numbers: \((x + a)(x + b) = 0\), where \(a\) and \(b\) are the numbers found in the previous step.
4. Set each binomial equal to zero to find the solutions: \(x + a = 0\) and \(x + b = 0\).
5. Solve each equation for \(x\) to find the roots of the quadratic.

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations involves rewriting the quadratic expression as a product of two binomials. This method is useful when the quadratic can be expressed as (x + a)(x + b) = 0, where a and b are numbers that multiply to the constant term and add to the coefficient of the linear term.

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 property allows us to set each factor equal to zero and solve for the variable, which is essential for finding the roots of the equation after factoring.

Solving Quadratic Equations

Solving quadratic equations means finding the values of the variable that satisfy the equation. After factoring, these solutions are found by setting each factor equal to zero and solving the resulting linear equations, yielding the roots or solutions of the quadratic.