Expand and simplify quadratic expressions

Ontario Grade 11 Mathematics

Mathematical Models

MBF3C, specific expectation A1.5

A quadratic expression is an expression in which the highest power of the variable is two. For example,

x^2+5x+6

is quadratic. Expanding means removing brackets by multiplying each term correctly. Simplifying means combining like terms so the expression is written more compactly. In this lesson, you will use multiplication and the distributive property to expand and simplify quadratic expressions. You will not solve equations; the goal is to rewrite expressions in an equivalent form.

What you will learn

Recognize a quadratic expression and identify its terms.
Use the distributive property to expand products that contain brackets.
Combine like terms to simplify an expanded expression.
Check an expansion by substituting a number for the variable.

1. Review the building blocks

A term is a part of an expression separated from other parts by a plus or minus sign. In

4x+7

, the terms are

4x

and

7

. The number multiplying a variable is its coefficient. So, the coefficient of

x

is

4

in

4x

. A constant is a number with no variable, such as

7

.

The exponent tells how many times a variable is used as a factor. Thus,

x^2

means

x

multiplied by

x

. It does not mean

2x

. In a quadratic expression, the

x^2

term is often called the squared term. For instance, in

3x^2-2x+8

, the squared term is

3x^2

, the linear term is

-2x

, and the constant is

8

.

Like terms have the same variable raised to the same power. You can combine

3x

and

5x

to get

8x

, but you cannot combine

x^2

with

x

. Their variable parts are different. This rule will matter after brackets are expanded.

x^2=x\cdot x

Keep squared terms, linear terms, and constants separate.
Only combine terms with matching variable parts.

2. Expand by multiplying every term

The distributive property says that a factor outside brackets multiplies every term inside. For example,

3(x+4)

becomes

3x+12

. The factor

3

must reach both

x

and

4

.

A product of two brackets needs the same care. In

(x+3)(x+2)

, each term in the first bracket multiplies each term in the second bracket. The four products are

x\cdot x

,

x\cdot2

,

3\cdot x

, and

3\cdot2

. This is sometimes remembered as first, outer, inner, last, but the important idea is to include every pair once.

The product

x\cdot x

gives

x^2

. The two middle products are both terms in

x

, so they can be combined. The final product is a constant. That pattern explains why multiplying two linear brackets can produce a quadratic expression.

A box or area model can help organize the products. Put the terms from one bracket along the top and the terms from the other bracket down the side. Multiply across each cell. This makes it easier to see whether any product has been missed.

(a+b)(c+d)=ac+ad+bc+bd

Multiply every term in one bracket by every term in the other bracket.
Write all products before combining like terms.

3. Simplify and check the result

After expansion, collect like terms. Put squared terms together, then linear terms, then constants. For example,

2x^2+3x+4x+1

simplifies to

2x^2+7x+1

because

3x

and

4x

are like terms.

Signs are part of terms. In

(x-4)(x+2)

, the first bracket contains

x

and

-4

. The product of

-4

and

2

is negative, and the product of

-4

and

x

is also negative. Keeping each sign attached to its term helps avoid errors.

You can check an expansion by using the same value of the variable in both forms. If the original and expanded expressions give different results for that value, an error has occurred. Matching values are a useful check, although it is still important to show the expansion steps.

A quadratic expression may describe a measurement, such as the area of a rectangle whose side lengths are expressions. If one side is

x+3

and the other is

x+2

, multiplying the side lengths gives an expression for the area. Expanding writes that area as a sum of a squared term, linear terms, and a constant. The expression is equivalent; its value does not change just because its form changes.

2x^2+3x+4x+1=2x^2+7x+1

Combine only like terms after all products are written.
Preserve negative signs during multiplication.
Substitution can help check that two forms are equivalent.

Area model for $(x+3)(x+2)$

Multiply by	$x$	$2$
$x$	$x^2$	$2x$
$3$	$3x$	$6$

Worked example

Example 1: Expand a product of two binomials

Expand and simplify

(x+4)(x+2)

.

List every product
Multiply each term in the first bracket by each term in the second. This ensures that all four pairings are included.
$x\cdot x+x\cdot2+4\cdot x+4\cdot2$
Multiply the terms
The variable multiplied by itself gives a squared term. The two middle products are terms in $x$ , and the last product is a constant.
$x^2+2x+4x+8$
Combine like terms
The terms $2x$ and $4x$ have the same variable part, so add their coefficients. Keep $x^2$ and $8$ separate.
$x^2+6x+8$

Answer: The expanded and simplified expression is

x^2+6x+8

.

Check: Substitute

x=1

. The original gives

(1+4)(1+2)=15

. The expanded form gives

1+6+8=15

, so the forms agree for this value.

Worked example

Example 2: Expand when a bracket contains a negative term

Expand and simplify

2(x-3)(x+5)

.

Expand the bracket product
First multiply the two bracketed expressions. Pair every term with every term, keeping the negative sign on $-3$ .
$(x-3)(x+5)=x^2+5x-3x-15$
Combine the linear terms
The terms $5x$ and $-3x$ are like terms. Adding their coefficients gives $2x$ .
$x^2+2x-15$
Multiply by the outside factor
The factor $2$ is outside the product, so it multiplies every term in the simplified expression.
$2x^2+4x-30$

Answer: The expanded and simplified expression is

2x^2+4x-30

.

Check: Substitute

x=2

. The original gives

2(2-3)(2+5)=-14

. The expanded form gives

2(4)+4(2)-30=-14

, so the forms agree for this value.

Common mistakes and how to avoid them

Multiplying only the first terms in the brackets and writing

x^2

as the whole answer.

Correction: Include all four pairings when multiplying two brackets. Then simplify the complete list of products.

Combining

x^2

and

x

because both contain the variable

x

.

Correction: They are not like terms: one has

x

squared and the other has

x

to the first power. Keep them as separate terms.

Losing a negative sign when multiplying a negative term.

Correction: Treat the negative sign as part of the term. For example,

-3

multiplied by

x

gives

-3x

.

Combining terms before all products have been accounted for.

Correction: First list and calculate every product. Then group and combine like terms.

Lesson summary

Expanding removes brackets by multiplying every required pair of terms.
A product of two brackets produces four products before simplification.
Combine only like terms, and keep signs attached to their terms.
Check an expanded form by substituting the same number into both forms.

Check your understanding

Question 1

Expand and simplify

(x+1)(x+6)

.

$x^2+7x+6$
$x^2+6x+1$
$2x+7$
$x^2+5x+6$

Show answer and explanation

x^2+7x+6

The four products are

x^2

,

6x

,

x

, and

6

. Combining the linear terms gives

x^2+7x+6

.

Question 2

Expand and simplify

(x-2)(x+3)

.

$x^2+x-6$
$x^2+5x-6$
$x^2-x+6$
$x^2+3x-2$

Show answer and explanation

x^2+x-6

The products are

x^2

,

3x

,

-2x

, and

-6

. Combining

3x

and

-2x

gives

x

.

Question 3

Expand and simplify

3(x+2)(x+1)

.

$3x^2+9x+6$
$3x^2+3x+6$
$3x^2+9x+2$
$x^2+9x+6$

Show answer and explanation

3x^2+9x+6

The bracket product is

x^2+3x+2

. Multiplying every term by

3

gives

3x^2+9x+6

.

Key terms

Quadratic expression: An expression whose highest variable power is two.
Expand: Remove brackets by multiplying the terms they contain.
Simplify: Rewrite an expression in a more compact form by combining like terms.
Like terms: Terms with the same variable part, including the same exponent.
Distributive property: A multiplication rule that requires a factor to multiply each term in brackets.

Continue through MBF3C

View the complete MBF3C Ontario Grade 11 Mathematics curriculum and lessons

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A1.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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