Represent situations with quadratic expressions and simplify them

Ontario Grade 11 Mathematics

Quadratic Functions

Build an expression from a situation, then simplify it

A situation can describe a quantity that changes. If the quantity depends on an unknown number, we can use a variable to represent it. When the situation involves multiplying two expressions that each contain the variable, the result may be quadratic. In this lesson, you will build such expressions from a situation and simplify them without changing their value.

What you will learn

Identify quantities that can be represented by a variable.
Write a quadratic expression to represent a situation.
Simplify a quadratic expression by expanding brackets and combining like terms.
Explain what the terms in a simplified expression represent.

1. Prerequisite bridge: variables and like terms

A variable is a letter used to represent a number that may be unknown or may change. For example,

x

could represent the side length of a shape. A numerical expression such as

x+4

represents a number that is four more than

x

.

A term is a number, a variable, or a product of numbers and variables. In

3x+7

, the terms are

3x

and

7

. Like terms have the same variable part. For example,

2x

and

5x

are like terms, but

2x

and

2x^2

are not.

You can combine like terms by adding or subtracting their number parts. So

2x+5x=7x

. The variable part stays the same. You cannot combine unlike terms, so

x^2+x

stays as it is.

An exponent tells how many times a base is used as a factor. Thus

x^2

means

x

multiplied by

x

. It does not mean

2x

. These facts help us read and simplify expressions that model area or other quantities.

2x+5x=7x

A variable represents a number.
Combine only like terms.
$x^2$ means $x$ multiplied by itself.

2. From a situation to a quadratic expression

A quadratic expression is an expression whose highest power of the variable is

2

. It can include a squared term, a term with the variable to the first power, and a constant. For example,

x^2+6x+8

is quadratic.

A useful way to represent a situation is to name the changing quantity first. Then write an expression for each quantity in the situation. Finally, use the operation that connects them. For area of a rectangle, multiply its length by its width.

Suppose a rectangular garden has a length that is three metres greater than its width. If the width is

x

metres, then its length is

x+3

metres. The area is the product of those two dimensions. This product is a quadratic expression because it includes multiplying two expressions that contain

x

.

The units also help make sense of the expression. A length is measured in metres, while area is measured in square metres. When the two dimensions are multiplied, the resulting expression represents area, not length.

A=x(x+3)

Name the unknown quantity and assign it a variable.
Translate each relationship in words into an expression.
Use the operation that matches the situation, such as multiplication for rectangular area.

3. Simplifying without changing the value

To simplify an expression, rewrite it in a shorter or more useful form while keeping the same value for every allowed value of the variable. One common step is expanding brackets. The distributive property says that a factor outside brackets multiplies every term inside them.

For example,

x(x+3)

means that

x

multiplies both

x

and

3

. This gives a squared term and a term with one

x

. If brackets contain two terms each, multiply each term in the first bracket by each term in the second bracket.

After expanding, combine like terms. For instance, if expansion produces

3x+2x

, combine those terms to get

5x

. Keep squared terms, first-power terms, and constants separate because they are not like terms.

A simplified quadratic expression is often written with the squared term first, then the first-power term, then the constant. This order makes the parts easier to identify. It does not change the value of the expression.

a(b+c)=ab+ac

Use the distributive property to expand brackets.
Multiply every term in one bracket by every term in the other bracket.
Combine like terms only after expanding.

4. Check that the expression fits the situation

A good model matches both the relationships and the units in the situation. If a variable represents a width in metres, then an expression such as

x+3

can represent a length in metres. Their product represents area in square metres.

You can also check a simplified expression by choosing a simple value for the variable and comparing it with the original expression. This is a numerical check, not a replacement for expanding correctly. If both forms give the same result, that supports your work.

Pay attention to the meaning of the variable. A width or length in a real situation is usually positive. You do not need to change the algebraic steps, but the situation may limit which values make sense.

When explaining your answer, state what the variable represents and what the whole expression represents. This makes the model clear to someone who has not seen the original situation.

Use units to check what the expression represents.
A numerical substitution can help check that two forms agree.
State the meaning of the variable and the expression.

How the patio expression is built

Part of the situation	Expression	Meaning
Width	$x+2$	$2$ metres more than $x$ metres
Length	$x+5$	$5$ metres more than $x$ metres
Area	$(x+2)(x+5)$	Length multiplied by width
Simplified area	$x^2+7x+10$	The same area written without brackets

Worked example

Area of a rectangular patio

A rectangular patio has a width of

x+2

metres and a length of

x+5

metres. Write an expression for its area and simplify it.

Name the quantities
The dimensions are already written in terms of $x$ . Area of a rectangle is length multiplied by width, so multiply the two dimension expressions.
$A=(x+2)(x+5)$
Expand the brackets
Multiply each term in the first bracket by each term in the second bracket. This applies the distributive property to all four pairs.
$(x+2)(x+5)=x^2+5x+2x+10$
Combine like terms
The terms $5x$ and $2x$ are like terms, so add their coefficients. The squared term and the constant are unlike those terms and stay separate.
$x^2+5x+2x+10=x^2+7x+10$

Answer: The patio's area is represented by

x^2+7x+10

square metres.

Check: If

x=1

, the original dimensions are

3

m and

6

m, so the area is

18

square metres. The simplified expression gives

1^2+7(1)+10=18

square metres.

Common mistakes and how to avoid them

Writing the area as

(x+2)+(x+5)

.

Correction: Area of a rectangle uses length multiplied by width, not added. Use

(x+2)(x+5)

.

Multiplying only the first terms when expanding brackets.

Correction: Multiply every term in one bracket by every term in the other bracket. Include all four products for two brackets with two terms each.

Combining

x^2

and

x

as if they were like terms.

Correction:

x^2

and

x

have different variable parts, so they cannot be combined.

Reading

x^2

as

2x

.

Correction:

x^2

means

x

multiplied by

x

. The expression

2x

means

2

multiplied by

x

.

Lesson summary

Choose a variable for the changing or unknown quantity.
Translate the situation into expressions and connect them with the correct operation.
For rectangular area, multiply length by width.
Expand brackets with the distributive property, then combine like terms.
Check that the expression and its units match the situation.

Check your understanding

Question 1

A rectangle has width

x+1

and length

x+4

. Which expression represents its area?

$(x+1)+(x+4)$
$(x+1)(x+4)$
$x+1+4$
correctIndexи 1

Show answer and explanation

(x+1)(x+4)

Area is length multiplied by width, so the dimensions are multiplied. The addition option gives a sum of lengths, not area.

Question 2

Simplify

x(x+6)

.

$x^2+6x$
$x^2+6$
$7x$
correctIndexи 0

Show answer and explanation

x^2+6x

Distribute

x

to both terms inside the bracket:

x\cdot x=x^2

and

x\cdot 6=6x

.

Question 3

Which pair contains like terms?

$4x$ and $4x^2$
$3x^2$ and $8x^2$
$x$ and $1$
correctIndexи 1

Show answer and explanation

3x^2

and

8x^2

Both terms in the correct pair have the same variable part,

x^2

. Their coefficients can be combined.

Key terms

Variable: A letter that represents a number that may be unknown or may change.
Term: A number, a variable, or a product of numbers and variables in an expression.
Like terms: Terms with the same variable part, such as $2x$ and $5x$ .
Quadratic expression: An expression whose highest power of its variable is $2$ .
Distributive property: A rule for multiplying a factor by every term inside brackets.
Simplify: Rewrite an expression in a shorter or more useful form without changing its value.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation A1.2. 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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