Answer each question. Find the values of a, b, and c for which the quadratic equation. ax^2 + bx + c = 0 has the given numbers as solutions. (Hint: Use the zero-factor property in reverse.) 4, 5

Verified step by step guidance

1. Start by using the fact that if 4 and 5 are solutions to the quadratic equation, then the equation can be expressed as (x - 4)(x - 5) = 0.
2. Expand the expression (x - 4)(x - 5) to get a quadratic equation in standard form.
3. Multiply the binomials: (x - 4)(x - 5) = x^2 - 5x - 4x + 20.
4. Combine like terms to simplify the expression: x^2 - 9x + 20.
5. Identify the coefficients a, b, and c from the quadratic equation x^2 - 9x + 20 = 0, where a = 1, b = -9, and c = 20.

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + 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, including factoring, completing the square, or applying the quadratic formula. Understanding the structure of quadratic equations is essential for solving them.

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 quadratic equations by factoring, as it allows us to set each factor equal to zero to find the solutions. In this context, it helps in determining the values of a, b, and c when given specific roots.

Factoring Quadratics

Factoring quadratics involves expressing the quadratic equation in the form (x - r1)(x - r2) = 0, where r1 and r2 are the roots of the equation. This method simplifies finding the coefficients a, b, and c by relating them to the roots. For example, if the roots are known, the equation can be expanded to identify the corresponding coefficients.