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

Verified step by step guidance

1. Recall that if a quadratic equation \(ax^2 + bx + c = 0\) has solutions (roots) \(r_1\) and \(r_2\), then it can be factored as \(a(x - r_1)(x - r_2) = 0\).
2. Given the solutions \$4\( and \)5\(, write the factored form of the quadratic as \)a(x - 4)(x - 5) = 0$.
3. Expand the factored form by first multiplying the binomials: \((x - 4)(x - 5) = x^2 - 5x - 4x + 20 = x^2 - 9x + 20\).
4. Multiply the expanded expression by \(a\) to get \(a x^2 - 9a x + 20a = 0\), which matches the general form \(ax^2 + bx + c = 0\).
5. Identify the coefficients: \(a\) is the leading coefficient, \(b = -9a\), and \(c = 20a\). You can choose any nonzero value for \(a\) to get specific values for \(b\) and \(c\).

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions (roots) of the equation are the values of x that satisfy it, often found using factoring, completing the square, or the quadratic formula.

Zero-Factor Property

The zero-factor property states that if the product of two factors equals zero, then at least one of the factors must be zero. This property is used to solve quadratic equations by factoring them into two binomials and setting each equal to zero to find the roots.

Forming Quadratic Equations from Roots

Given the roots of a quadratic equation, you can construct the equation by reversing the factoring process. If the roots are r₁ and r₂, the quadratic can be written as a(x - r₁)(x - r₂) = 0, which expands to ax² - a(r₁ + r₂)x + a(r₁r₂) = 0, allowing you to identify a, b, and c.