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Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Expanding and Simplifying Second-Degree Expressions

3.1About 70 minutes
Learning goals
  • I can expand and simplify second-degree polynomial expressions, including ones with two variables.
  • I can use area models and algebra tiles to show why a product expands the way it does.
  • I can use the patterns (a + b)², (a − b)² and (a + b)(a − b) to expand quickly.
  • I can check an expansion by substituting a number.

Words to know

second-degree polynomial
A polynomial whose highest-degree term has degree 2, such as 3x² − 5x + 1 or x² + xy. Also called a quadratic expression.
degree of a term
The sum of the exponents on the variables in the term. The term 4xy has degree 2.
expand
Multiply out the brackets in an expression.
simplify
Collect like terms so an expression has as few terms as possible.
binomial
A polynomial with two terms, such as 2x − 3.
trinomial
A polynomial with three terms, such as x² + 5x + 6.
perfect square
The product of a binomial with itself, such as (x + 6)² = x² + 12x + 36.
difference of squares
An expression of the form a² − b². It comes from (a + b)(a − b).

A mental-math trick with a secret

Can you work out 47 × 53 in your head in a few seconds? Here is how a mathematician does it.

47 is 50 − 3 and 53 is 50 + 3. So 47 × 53 = (50 − 3)(50 + 3) = 50² − 3² = 2500 − 9 = 2491.

That trick is really algebra. It works because (a − b)(a + b) = a² − b² for any numbers a and b. In this unit you will use patterns like this one again and again. They are the key to factoring and to solving quadratic equations, which is where this unit is heading.

Remember from Grade 9

In Grade 9 you used the distributive property, collected like terms, and multiplied two binomials with algebra tiles and area models, for example (x + 4)(x − 1) = x² + 3x − 4. You also saw that (x + 5)² = x² + 10x + 25, not x² + 25, and that (x + 3)(x − 3) = x² − 9.

In the unit Quadratic Relations and Their Graphs you met relations such as y = x² + 2x − 3 and y = 2(x − 1)² + 4. This unit gives you the algebra to move between those forms.

What is a second-degree polynomial?

The degree of a term is the sum of the exponents on its variables. For example, 5x² has degree 2, and 3xy also has degree 2 (because x¹y¹ gives 1 + 1 = 2).

A second-degree polynomial (also called a quadratic expression) has degree 2: its highest-degree term has degree 2. Examples: x² − 7x + 10, 4x² − 25, 2x² + 5xy − 3y².

Expanding means multiplying out the brackets. Simplifying means collecting like terms so the expression is as short as possible. The rule is always the same: multiply every term in one bracket by every term in the other bracket, then collect like terms.

Worked example

Example 1: Square a binomial with a coefficient

Expand and simplify (2x + 5)².

  1. (2x + 5)² means (2x + 5)(2x + 5). Never square each term on its own.
  2. Area model: 2x × 2x = 4x²; 2x × 5 = 10x; 5 × 2x = 10x; 5 × 5 = 25.
  3. Add the four boxes: 4x² + 10x + 10x + 25.
  4. Collect like terms: 10x + 10x = 20x.
  5. Check with x = 1: (2 + 5)² = 49, and 4 + 20 + 25 = 49 ✓.

(2x + 5)² = 4x² + 20x + 25

2x+52x4x²10x+510x25Add the boxes:4x² + 10x + 10x + 25= 4x² + 20x + 25
An area model for (2x + 5)². Notice the two 10x boxes: they become the middle term 20x.
Worked example

Example 2: Two variables

Expand and simplify (2x − y)(x + 3y).

  1. Multiply each term of the first bracket by each term of the second.
  2. 2x × x = 2x²; 2x × 3y = 6xy; (−y) × x = −xy; (−y) × 3y = −3y².
  3. Add: 2x² + 6xy − xy − 3y².
  4. Like terms: 6xy − xy = 5xy. (xy and yx are like terms.)
  5. Check with x = 1, y = 1: (2 − 1)(1 + 3) = 4, and 2 + 5 − 3 = 4 ✓.

2x² + 5xy − 3y²

x+3y2x2x²6xy−y−xy−3y²Collect the xy-terms:6xy − xy = 5xy2x² + 5xy − 3y²
The same rule works with two variables. Keep the sign with each term: the side term is −y, so its row of products is negative.

Three patterns worth memorizing

Some products appear so often that it pays to know them by heart. Each one comes from the same "every term times every term" rule.

PatternRuleExample
Square of a sum(a + b)² = a² + 2ab + b²(x + 6)² = x² + 12x + 36
Square of a difference(a − b)² = a² − 2ab + b²(3x − 4)² = 9x² − 24x + 16
Product of a sum and a difference(a + b)(a − b) = a² − b²(2x + 7)(2x − 7) = 4x² − 49
In a perfect square the middle term is 2 × first × last. In a sum-times-difference product the middle terms cancel.
a²ababb²ababSide length: a + bArea = (a + b)²= a² + ab + ab + b²= a² + 2ab + b²The two ab rectangles arewhy the middle term is 2ab.
A square with side a + b splits into a², two ab rectangles and b². This picture is why (a + b)² = a² + 2ab + b², and why (a + b)² is not a² + b².
Worked example

Example 3: Simplify a longer expression

Expand and simplify 3(x − 2)² − (2x + 1)(2x − 1).

  1. Square first (order of operations): (x − 2)² = x² − 4x + 4.
  2. Multiply by 3: 3x² − 12x + 12.
  3. Sum times difference: (2x + 1)(2x − 1) = 4x² − 1.
  4. Subtract the whole product. Keep it in brackets: 3x² − 12x + 12 − (4x² − 1).
  5. Distribute the negative: 3x² − 12x + 12 − 4x² + 1.
  6. Collect: (3x² − 4x²) − 12x + (12 + 1) = −x² − 12x + 13.
  7. Check with x = 0: 3(4) − (1)(−1) = 12 + 1 = 13, and the answer gives 13 ✓.

−x² − 12x + 13

Worked example

Example 4: An area in context

A rectangular poster for a winter carnival is (3x + 2) cm long and (2x − 1) cm wide. Write a simplified expression for its area, then find the area when x = 10.

  1. Area = length × width = (3x + 2)(2x − 1).
  2. Expand: 6x² − 3x + 4x − 2.
  3. Collect: 6x² + x − 2.
  4. When x = 10: 6(100) + 10 − 2 = 608.
  5. Check: the poster is 32 cm by 19 cm, and 32 × 19 = 608 ✓.

Area = 6x² + x − 2 cm²; when x = 10 the area is 608 cm².

Watch out

  • (a + b)² ≠ a² + b². Write the square as two brackets if you need to.
  • When you subtract a product, put the expanded product in brackets first, then change every sign inside.
  • Square coefficients too: (3x)² = 9x², not 3x².
  • Always check by substituting a simple number such as x = 1 or x = 0 into the question and your answer.
Hands-on activity

Prove the square pattern on grid paper

You need: grid paper (or draw your own grid), a pencil, two coloured pencils, a calculator.

  1. Draw a square that is 13 squares by 13 squares.
  2. Draw one line across and one line down so the sides are split into 10 + 3.
  3. Colour the 10-by-10 square, the two 10-by-3 rectangles and the 3-by-3 square differently.
  4. Count or multiply: 100 + 30 + 30 + 9 = 169. Check 13² = 169 on the calculator.
  5. Write the algebra: (10 + 3)² = 10² + 2(10)(3) + 3².
  6. Now use the pattern in your head: 21², 32² and 99² (use 100 − 1). Check each on the calculator.
  7. Use the sum-and-difference trick for 38 × 42 and 59 × 61.

Model answers: 21² = 400 + 40 + 1 = 441; 32² = 900 + 120 + 4 = 1024; 99² = 10 000 − 200 + 1 = 9801; 38 × 42 = 1600 − 4 = 1596; 59 × 61 = 3600 − 1 = 3599.

Just for fun

Why did the binomial bring two brackets to the party?

It wanted to be expanded to its full potential. (Sadly, its middle term was so outgoing that it ended up counting twice.)

Your turn

1

Expand and simplify (x + 7)(x − 3).

2

Expand and simplify (3x + 2)(x + 4).

3

Expand (2x − 3)².

4

Expand (5x + 4)(5x − 4).

5

Expand and simplify (x + 2y)(3x − y).

6

Expand and simplify 2(x + 1)(x − 5).

7

Expand and simplify (x + 4)² − (x − 4)².

More practice: warm-up

1

Expand (x + 3)(x + 5).

2

Expand (x − 6)(x − 1).

3

Expand (x + 9)².

4

Expand (x − 10)(x + 10).

5

Expand 4x(x − 3).

More practice: on your own

1

Expand and simplify (4x − 1)(2x + 5).

2

Expand (3a − 2b)².

3

Expand and simplify 3(2x − 1)² − 5x(x + 2).

4

A rectangular deck is (2x + 3) m by (x + 4) m. A square hot tub with side x m is set into the deck. Write a simplified expression for the deck area that is left, and evaluate it for x = 2.

5

Use a product pattern to calculate 98 × 102 without a calculator.

Challenge

1

If (ax + 3)² = 16x² + bx + 9 and a > 0, find a and b.

2

Expand and simplify (x + 2y − 1)(x − 2y + 1). (Hint: group the terms after x.)

3

Show that if you multiply two consecutive odd numbers and add 1, you always get a perfect square. (Let the odd numbers be 2n − 1 and 2n + 1.) Test your result with 5 and 7.

Check your understanding

Quiz

1.

(x − 5)² is equal to:

2.

Expand and simplify (2x + 3)(x − 4).

3.

(3x + 7)(3x − 7) simplifies to:

4.

What is the xy-term when (x + 6y)(x − 2y) is expanded and simplified?

5.

Which expression is a second-degree polynomial?

6.

Simplify (x + 1)² − (x − 1)².

For parents and teachers

Big idea: Expectation 3.1 asks students to expand and simplify second-degree expressions such as (2x + 5)² and (2x − y)(x + 3y) using tools such as algebra tiles, diagrams and patterning. Every later lesson in this unit (factoring, completing the square, the quadratic formula) depends on this skill, so fluency here pays off.

How to help: Ask your teen to show you an area model for one question. Then pick a number for x (such as 2) and check that the question and the answer give the same value. Mental-math tricks such as 47 × 53 = 50² − 3² are a fun way to practise the patterns at the dinner table.

Common difficulties:

  • Writing (x + 5)² as x² + 25 (forgetting the middle term).
  • Sign errors when subtracting a product: −(4x² − 1) must become −4x² + 1.
  • Forgetting to square the coefficient: (3x)² = 9x².
  • Treating xy and x² as like terms. They are not.