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

Lesson 3: Factoring Quadratics II

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to factor quadratic expressions of the form ax² + bx + c where the leading coefficient a is not equal to 1. Building on binomial multiplication, students apply number sense strategies including parity analysis and coefficient magnitude to systematically identify the correct binomial factorizations. The lesson covers key techniques for matching factors of a and c using the linear term b as a guide, with worked examples such as factoring 5x² − 36x + 7 and 8x² + 23x + 15.

Section 1

Clearing Fractions Before Factoring

Property

When a quadratic expression contains fractions, multiply all terms by the least common denominator (LCD) to clear fractions before factoring:

\frac{a}{d}x^2 + \frac{b}{e}x + \frac{c}{f} \rightarrow \text{LCD} \cdot \left(\frac{a}{d}x^2 + \frac{b}{e}x + \frac{c}{f}\right)

Examples

Section 2

Factoring Out Common Factors First

Property

Before factoring a quadratic expression $ax^2 + bx + c$ , always check for and factor out the greatest common factor (GCF) of all terms first: $k(a'x^2 + b'x + c')$ where $k$ is the GCF.

Examples

Section 3

Multiplying binomials

Property

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. The acronym FOIL helps organize the four products:

First terms
Outer terms
Inner terms
Last terms

(x-4)(x+6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24

Examples

Using FOIL for $(x+2)(x+7)$ : $x^2$ (F) $+ 7x$ (O) $+ 2x$ (I) $+ 14$ (L), which simplifies to $x^2 + 9x + 14$ .

For $(3y-2)(y+5)$ : $3y^2$ (F) $+ 15y$ (O) $- 2y$ (I) $- 10$ (L), which simplifies to $3y^2 + 13y - 10$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Clearing Fractions Before Factoring

Property

When a quadratic expression contains fractions, multiply all terms by the least common denominator (LCD) to clear fractions before factoring:

\frac{a}{d}x^2 + \frac{b}{e}x + \frac{c}{f} \rightarrow \text{LCD} \cdot \left(\frac{a}{d}x^2 + \frac{b}{e}x + \frac{c}{f}\right)

Examples

Section 2

Factoring Out Common Factors First

Property

Before factoring a quadratic expression $ax^2 + bx + c$ , always check for and factor out the greatest common factor (GCF) of all terms first: $k(a'x^2 + b'x + c')$ where $k$ is the GCF.

Examples

Section 3

Multiplying binomials

Property

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. The acronym FOIL helps organize the four products:

First terms
Outer terms
Inner terms
Last terms

(x-4)(x+6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24

Examples

Using FOIL for $(x+2)(x+7)$ : $x^2$ (F) $+ 7x$ (O) $+ 2x$ (I) $+ 14$ (L), which simplifies to $x^2 + 9x + 14$ .

For $(3y-2)(y+5)$ : $3y^2$ (F) $+ 15y$ (O) $- 2y$ (I) $- 10$ (L), which simplifies to $3y^2 + 13y - 10$ .

Overview

Lesson overview

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

Overview

Section 1

Clearing Fractions Before Factoring

Property

When a quadratic expression contains fractions, multiply all terms by the least common denominator (LCD) to clear fractions before factoring:

\frac{a}{d}x^2 + \frac{b}{e}x + \frac{c}{f} \rightarrow \text{LCD} \cdot \left(\frac{a}{d}x^2 + \frac{b}{e}x + \frac{c}{f}\right)

Examples

Section 2

Factoring Out Common Factors First

Property

Before factoring a quadratic expression $ax^2 + bx + c$ , always check for and factor out the greatest common factor (GCF) of all terms first: $k(a'x^2 + b'x + c')$ where $k$ is the GCF.

Examples

Section 3

Multiplying binomials

Property

To multiply two binomials, multiply each term of the first binomial by each term of the second binomial. The acronym FOIL helps organize the four products:

First terms
Outer terms
Inner terms
Last terms

(x-4)(x+6) = x^2 + 6x - 4x - 24 = x^2 + 2x - 24

Examples

Using FOIL for $(x+2)(x+7)$ : $x^2$ (F) $+ 7x$ (O) $+ 2x$ (I) $+ 14$ (L), which simplifies to $x^2 + 9x + 14$ .

For $(3y-2)(y+5)$ : $3y^2$ (F) $+ 15y$ (O) $- 2y$ (I) $- 10$ (L), which simplifies to $3y^2 + 13y - 10$ .