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

Lesson 2: Factoring Quadratics I

In this Grade 4 AoPS Introduction to Algebra lesson, students learn how to factor quadratic expressions of the form x² + bx + c by finding two numbers r and s such that r + s equals the linear coefficient and rs equals the constant term. The lesson covers identifying roots and zeros of a quadratic equation by rewriting it as a product of binomials and applying the zero-product property. Part of Chapter 10 on Quadratic Equations, this lesson builds problem-solving strategies for AMC 8 and AMC 10 competition math.

Section 1

Factor Quadratics with c = 0

Property

When $c = 0$ , the quadratic $x^2 + bx$ can be factored by taking out the common factor $x$ :

x^2 + bx = x(x + b)

Section 2

Factor Trinomials of the Form $x^2 + bx + c$

Property

To factor a trinomial of the form $x^2 + bx + c$ , we need two factors $(x + m)$ and $(x + n)$ where the two numbers $m$ and $n$ multiply to $c$ and add to $b$ .

How to factor trinomials of the form $x^2 + bx + c$

Write the factors as two binomials with first terms $x$ : $(x \quad)(x \quad)$ .
Find two numbers $m$ and $n$ that multiply to $c$ , $m \cdot n = c$ , and add to $b$ , $m + n = b$ .
Use $m$ and $n$ as the last terms of the factors.
Check by multiplying the factors.

Strategy for Determining Signs
When $c$ is positive, $m$ and $n$ have the same sign as $b$ .

If $b$ is positive, $m$ and $n$ are positive. Example: $x^2 + 5x + 6 = (x+2)(x+3)$ .
If $b$ is negative, $m$ and $n$ are negative. Example: $x^2 - 6x + 8 = (x-4)(x-2)$ .

Section 3

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

a^2 + 2ab + b^2 = (a+b)^2

a^2 - 2ab + b^2 = (a-b)^2

To use this pattern, first verify that the trinomial fits.
Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

).
Then, check if the middle term is twice their product (

2ab

).
If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Factor Quadratics with c = 0

Property

When $c = 0$ , the quadratic $x^2 + bx$ can be factored by taking out the common factor $x$ :

x^2 + bx = x(x + b)

Section 2

Factor Trinomials of the Form $x^2 + bx + c$

Property

To factor a trinomial of the form $x^2 + bx + c$ , we need two factors $(x + m)$ and $(x + n)$ where the two numbers $m$ and $n$ multiply to $c$ and add to $b$ .

How to factor trinomials of the form $x^2 + bx + c$

Write the factors as two binomials with first terms $x$ : $(x \quad)(x \quad)$ .
Find two numbers $m$ and $n$ that multiply to $c$ , $m \cdot n = c$ , and add to $b$ , $m + n = b$ .
Use $m$ and $n$ as the last terms of the factors.
Check by multiplying the factors.

Strategy for Determining Signs
When $c$ is positive, $m$ and $n$ have the same sign as $b$ .

If $b$ is positive, $m$ and $n$ are positive. Example: $x^2 + 5x + 6 = (x+2)(x+3)$ .
If $b$ is negative, $m$ and $n$ are negative. Example: $x^2 - 6x + 8 = (x-4)(x-2)$ .

Section 3

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

a^2 + 2ab + b^2 = (a+b)^2

a^2 - 2ab + b^2 = (a-b)^2

To use this pattern, first verify that the trinomial fits.
Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

).
Then, check if the middle term is twice their product (

2ab

).
If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Overview

Lesson overview

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

Overview

Section 1

Factor Quadratics with c = 0

Property

When $c = 0$ , the quadratic $x^2 + bx$ can be factored by taking out the common factor $x$ :

x^2 + bx = x(x + b)

Section 2

Factor Trinomials of the Form $x^2 + bx + c$

Property

To factor a trinomial of the form $x^2 + bx + c$ , we need two factors $(x + m)$ and $(x + n)$ where the two numbers $m$ and $n$ multiply to $c$ and add to $b$ .

How to factor trinomials of the form $x^2 + bx + c$

Write the factors as two binomials with first terms $x$ : $(x \quad)(x \quad)$ .
Find two numbers $m$ and $n$ that multiply to $c$ , $m \cdot n = c$ , and add to $b$ , $m + n = b$ .
Use $m$ and $n$ as the last terms of the factors.
Check by multiplying the factors.

Strategy for Determining Signs
When $c$ is positive, $m$ and $n$ have the same sign as $b$ .

If $b$ is positive, $m$ and $n$ are positive. Example: $x^2 + 5x + 6 = (x+2)(x+3)$ .
If $b$ is negative, $m$ and $n$ are negative. Example: $x^2 - 6x + 8 = (x-4)(x-2)$ .

Section 3

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

a^2 + 2ab + b^2 = (a+b)^2

a^2 - 2ab + b^2 = (a-b)^2

To use this pattern, first verify that the trinomial fits.
Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

).
Then, check if the middle term is twice their product (

2ab

).
If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .