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

Verified step by step guidance

1. Start by recalling that if the solution set of a quadratic equation is given as {x₁, x₂}, the equation can be written in factored form as (x - x₁)(x - x₂) = 0. Here, the solutions are x₁ = -3 and x₂ = 5.
2. Substitute the given solutions into the factored form: (x - (-3))(x - 5) = 0. Simplify the double negative to get (x + 3)(x - 5) = 0.
3. Expand the factored form using the distributive property: (x + 3)(x - 5) = x² - 5x + 3x - 15.
4. Combine like terms to simplify the expanded expression: x² - 5x + 3x - 15 = x² - 2x - 15.
5. Write the quadratic equation in general form: x² - 2x - 15 = 0. This is the required equation whose solution set is {-3, 5}.

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

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

The general form of a quadratic equation is written as ax² + bx + c = 0. To convert the roots into this form, one can use the factored form derived from the roots, which is a(x + 3)(x - 5). Expanding this expression will yield the general form, allowing for the identification of coefficients a, b, and c.