Grade by Grade
A. Quadratic Functions · Quadratic Expressions and Equations

Expanding and Simplifying Quadratic Expressions

A1.2About 65 minutes
Learning goals
  • I can expand products of binomials using the distributive property or an area model.
  • I can use the special products (a ± b)² and (a + b)(a − b).
  • I can expand and simplify expressions with several products, including subtracted products and fraction or decimal coefficients.

Words to know

expand
Rewrite a product as a sum of terms using the distributive property.
simplify
Collect like terms so that the expression has as few terms as possible.
like terms
Terms with the same variable raised to the same power, such as 3x and −7x.
binomial
A polynomial with two terms, such as 2x − 5.
trinomial
A polynomial with three terms, such as x² + 5x + 6.
standard form
A quadratic expression written as ax² + bx + c, with a ≠ 0.

Why expand?

In the last lesson you answered questions from tables and graphs. To get exact answers you need to work with the equations, and equations from real situations rarely arrive in a tidy form. A garden with a path, a frame around a photo or a price change all produce products like (x + 3)(2x − 1) or (x + 4)². Expanding rewrites a product as a sum of terms. Simplifying collects like terms so the expression is as short as possible. A simplified quadratic expression has the standard form ax² + bx + c.

Remember from Grade 10

You expanded binomial products with the distributive property, area models and algebra tiles, and you met the special products (a + b)², (a − b)² and (a + b)(a − b). This lesson practises all of these at a faster pace and adds longer expressions with several products, subtraction of whole products, and fraction and decimal coefficients.

The distributive property does everything

Every expansion is the distributive property: multiply each term in the first bracket by each term in the second, then collect like terms.

For two binomials there are four products. An area model keeps track of them: the rectangle's sides are the two binomials, and each small rectangle is one product. Signs travel with their terms.

(3x − 2)(2x + 5)3x−22x6x²−4x+515x−106x² − 4x + 15x − 10= 6x² + 11x − 10
An area model for (3x − 2)(2x + 5). The four small areas are the four products. The two x-terms, −4x and 15x, are like terms, so they combine to 11x.
Worked example

Example 1: Two binomials

Expand and simplify (3x − 2)(2x + 5).

  1. Multiply each term of the first bracket by each term of the second:
  2. 3x(2x) = 6x², 3x(5) = 15x, −2(2x) = −4x, −2(5) = −10.
  3. Collect like terms: 15x − 4x = 11x.
  4. Check with a number: let x = 1. Original: (1)(7) = 7. Answer: 6 + 11 − 10 = 7 ✓.

6x² + 11x − 10

Special products: patterns that save time

  • Square of a sum: (a + b)² = a² + 2ab + b²
  • Square of a difference: (a − b)² = a² − 2ab + b²
  • Product of a sum and a difference: (a + b)(a − b) = a² − b²

The most common mistake in the whole unit is writing (x + 3)² = x² + 9. The square below shows why that is wrong: a square of side a + b has four parts, not two.

ababa²ababb²(a + b)²= a² + ab + ab + b²= a² + 2ab + b²The middle term isTWICE the product ab.
A square with side a + b is made of a², b² and two rectangles of area ab. So (a + b)² = a² + 2ab + b², never just a² + b².
Worked example

Example 2: Using the patterns

Expand: (a) (2x − 3)² (b) (5x + 4)(5x − 4)

  1. (a) Use (a − b)² = a² − 2ab + b² with a = 2x and b = 3.
  2. a² = (2x)² = 4x²; 2ab = 2(2x)(3) = 12x; b² = 9. So (2x − 3)² = 4x² − 12x + 9.
  3. (b) Use (a + b)(a − b) = a² − b² with a = 5x and b = 4.
  4. (5x)² − 4² = 25x² − 16. The middle terms +20x and −20x cancel, which is why there is no x-term.

(a) 4x² − 12x + 9 (b) 25x² − 16

Worked example

Example 3: A longer expression

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

  1. Expand each part separately, and keep the second product in brackets because the whole thing is being subtracted.
  2. 2x(x + 4) = 2x² + 8x.
  3. (x + 3)² = x² + 6x + 9, so we subtract (x² + 6x + 9).
  4. 2x² + 8x − (x² + 6x + 9) = 2x² + 8x − x² − 6x − 9. The minus sign changes every sign inside the bracket.
  5. Collect: (2x² − x²) + (8x − 6x) − 9 = x² + 2x − 9.
  6. Check with x = 1: original 2(5) − 16 = −6; answer 1 + 2 − 9 = −6 ✓.

x² + 2x − 9

Worked example

Example 4: Fractions and decimals

Expand and simplify (a) (0.5x + 3)(x − 4) (b) 3(x − 2)² − (2x + 1)(x − 5)

  1. (a) 0.5x(x) = 0.5x², 0.5x(−4) = −2x, 3(x) = 3x, 3(−4) = −12. Collect: 0.5x² + x − 12.
  2. (b) Square first, then multiply by 3: (x − 2)² = x² − 4x + 4, so 3(x − 2)² = 3x² − 12x + 12.
  3. (2x + 1)(x − 5) = 2x² − 10x + x − 5 = 2x² − 9x − 5.
  4. Subtract the whole product: 3x² − 12x + 12 − 2x² + 9x + 5 = x² − 3x + 17.

(a) 0.5x² + x − 12 (b) x² − 3x + 17

Order of operations still rules

In 3(x − 2)², the square comes first, then the multiplication by 3. Do not write (3x − 6)²: that equals 9(x − 2)², which is three times too big. And when a product is subtracted, keep it in brackets until you have distributed the minus sign.

Just for fun

A student wrote (x + 3)² = x² + 9 on a test. The teacher wrote back: "You've lost your middle term. It was last seen as 6x, heading for the margins."

Hands-on activity

Check expansions with a paper square

You need: a sheet of paper, a ruler, a pencil and scissors (take care with scissors).

  1. Draw a square 13 cm by 13 cm. Mark a point 10 cm along the top edge and a point 10 cm down the left edge. Draw lines across to split the square into four rectangles: 10 × 10, 10 × 3, 3 × 10 and 3 × 3.
  2. Label the areas and add them. Compare with 13² = 169.
  3. This is (10 + 3)² = 10² + 2(10)(3) + 3². Explain in one sentence why 10² + 3² = 109 is not the area.
  4. Cut out the 3 × 3 square and the two 10 × 3 strips. Use them to show a friend or family member where the "2ab" comes from.

What to look for: 100 + 30 + 30 + 9 = 169. The two 10 × 3 strips (60 cm²) are exactly the middle term 2ab that is missing from 10² + 3².

Your turn

1

Expand and simplify (x + 7)(x − 2).

2

Expand (x − 6)².

3

Expand and simplify (4x + 1)(x − 3).

4

Expand (3x + 5)².

5

Expand (x − 9)(x + 9).

6

Expand and simplify −2(x + 3)(x − 1).

7

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

More practice: warm-up

1

Expand 3x(x − 5).

2

Expand (x + 2)(x + 5).

3

Expand (x + 4)².

4

Expand (x − 1)(x + 1).

5

Simplify x(x + 5) + 3(x − 2).

More practice: on your own

1

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

2

Expand (x/2 + 1)².

3

Expand and simplify −(3x − 1)² + 4x(x + 2).

4

Expand and simplify (1.5x − 2)(2x + 4).

5

Lee says (2x − 3)² = 4x² − 9. Find the error and correct it.

Challenge

1

Simplify 5(x + 1)² − 5(x − 1)².

2

Show that (x + 4)² − (x − 4)² = 16x for every x. Then use it to work out 104² − 96² in your head.

3

Expand and simplify (2x − 1)(x + 3) − 2(x + 1)². Then find the value of the expression when x = 10 in two ways.

Check your understanding

Quiz

1.

Which is the expansion of (x + 5)²?

2.

Expand (2x − 1)(x + 4).

3.

(3x + 2)(3x − 2) equals:

4.

Simplify x(x + 2) − (x + 1)².

5.

In 2(x − 3)², what is done first?

6.

The standard form of a quadratic expression is:

For parents and teachers

Big idea (A1.2, expanding and simplifying): every expansion is the distributive property. Special products are shortcuts, not new rules.

How to help: ask your teen to check each answer by substituting a number (x = 1 or x = 2) into the original and the answer. If the two values differ, there is an error. This one habit catches most mistakes in this unit.

Common difficulties:

  • Writing (a + b)² = a² + b². The paper-square activity shows why that is wrong.
  • Forgetting to change every sign when subtracting a product: −(x² + 6x + 9) = −x² − 6x − 9.
  • Multiplying by a coefficient before squaring: 3(x − 2)² is not (3x − 6)².
  • Sign errors with negative terms. Encourage writing each of the four products separately before collecting.