Write a quadratic equation in general form whose solution set is {- 3, 5}.

Verified step by step guidance

1. Start with the fact that if a quadratic equation has solutions \(x = a\) and \(x = b\), it can be expressed in factored form as \((x - a)(x - b) = 0\).
2. Substitute the given solutions into the factored form: \((x + 3)(x - 5) = 0\).
3. Expand the factored form by using the distributive property (also known as FOIL for binomials): \(x(x - 5) + 3(x - 5)\).
4. Simplify the expression by distributing: \(x^2 - 5x + 3x - 15\).
5. Combine like terms to write the quadratic equation in general form: \(x^2 - 2x - 15 = 0\).

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of degree two, typically expressed in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to this equation, known as the roots, can be found using various methods such as factoring, completing the square, or the quadratic formula.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of x that satisfy the equation, meaning they make the equation equal to zero. For a quadratic with roots r₁ and r₂, the equation can be expressed in factored form as a(x - r₁)(x - r₂) = 0. In this case, the roots are given as -3 and 5.

General Form of a Quadratic Equation

The general form of a quadratic equation is represented as ax² + bx + c = 0. To write a quadratic equation in this form given its roots, one can use the factored form and expand it. For roots -3 and 5, the equation can be derived by multiplying the factors (x + 3)(x - 5) and then rearranging it into the general form.