Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 10: Quadratic Equations - Part 1

Lesson 4: Sums and Products of Roots of a Quadratic

In this lesson from AoPS: Introduction to Algebra, Grade 4 students learn how to use Vieta's formulas to find the sum and product of the roots of a quadratic equation ax² + bx + c = 0, where the sum of the roots equals −b/a and the product equals c/a. The lesson derives these relationships by expanding the factored form a(x − r)(x − s) and equating coefficients, then applies them to solve problems involving unknown coefficients. Students practice multiple solution strategies, including substitution, factored-form construction, and direct use of the root-coefficient relationships.

Section 1

Deriving Sum and Product Formulas for Quadratic Roots

Property

For a quadratic equation $ax^2 + bx + c = 0$ with roots $r$ and $s$ :

\text{Sum of roots} = r + s = -\frac{b}{a}

\text{Product of roots} = rs = \frac{c}{a}

Examples

Section 2

Identifying Monic vs Non-Monic Quadratics for Root Formulas

Property

For a quadratic $ax^2 + bx + c = 0$ :

If $a = 1$ (monic): sum = $-b$ , product = $c$
If $a \neq 1$ (non-monic): sum = $-\frac{b}{a}$ , product = $\frac{c}{a}$

Examples

Section 3

Factored Form of an Equation

Property

The solutions of the quadratic equation $a(x - r_1)(x - r_2) = 0$ are $r_1$ and $r_2$ . This is called the factored form of the quadratic equation. If you know the two solutions of a quadratic equation, you can work backwards to reconstruct the equation.

Examples

A quadratic equation has solutions $x=3$ and $x=-6$ . The factors are $(x-3)$ and $(x-(-6))$ , or $(x+6)$ . The equation is $(x-3)(x+6)=0$ , which expands to $x^2+3x-18=0$ .

To find an equation with solutions $x=2$ and $x=\frac{1}{4}$ , start with factors $(x-2)$ and $(x-\frac{1}{4})$ . For integer coefficients, use $(x-2)$ and $(4x-1)$ . The equation is $(x-2)(4x-1)=0$ , or $4x^2-9x+2=0$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Deriving Sum and Product Formulas for Quadratic Roots

Property

For a quadratic equation $ax^2 + bx + c = 0$ with roots $r$ and $s$ :

\text{Sum of roots} = r + s = -\frac{b}{a}

\text{Product of roots} = rs = \frac{c}{a}

Examples

Section 2

Identifying Monic vs Non-Monic Quadratics for Root Formulas

Property

For a quadratic $ax^2 + bx + c = 0$ :

If $a = 1$ (monic): sum = $-b$ , product = $c$
If $a \neq 1$ (non-monic): sum = $-\frac{b}{a}$ , product = $\frac{c}{a}$

Examples

Section 3

Factored Form of an Equation

Property

Examples

A quadratic equation has solutions $x=3$ and $x=-6$ . The factors are $(x-3)$ and $(x-(-6))$ , or $(x+6)$ . The equation is $(x-3)(x+6)=0$ , which expands to $x^2+3x-18=0$ .

To find an equation with solutions $x=2$ and $x=\frac{1}{4}$ , start with factors $(x-2)$ and $(x-\frac{1}{4})$ . For integer coefficients, use $(x-2)$ and $(4x-1)$ . The equation is $(x-2)(4x-1)=0$ , or $4x^2-9x+2=0$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Deriving Sum and Product Formulas for Quadratic Roots

Property

For a quadratic equation $ax^2 + bx + c = 0$ with roots $r$ and $s$ :

\text{Sum of roots} = r + s = -\frac{b}{a}

\text{Product of roots} = rs = \frac{c}{a}

Examples

Section 2

Identifying Monic vs Non-Monic Quadratics for Root Formulas

Property

For a quadratic $ax^2 + bx + c = 0$ :

If $a = 1$ (monic): sum = $-b$ , product = $c$
If $a \neq 1$ (non-monic): sum = $-\frac{b}{a}$ , product = $\frac{c}{a}$

Examples

Section 3

Factored Form of an Equation

Property

Examples

A quadratic equation has solutions $x=3$ and $x=-6$ . The factors are $(x-3)$ and $(x-(-6))$ , or $(x+6)$ . The equation is $(x-3)(x+6)=0$ , which expands to $x^2+3x-18=0$ .

To find an equation with solutions $x=2$ and $x=\frac{1}{4}$ , start with factors $(x-2)$ and $(x-\frac{1}{4})$ . For integer coefficients, use $(x-2)$ and $(4x-1)$ . The equation is $(x-2)(4x-1)=0$ , or $4x^2-9x+2=0$ .