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A. Quadratic Functions · Quadratic Expressions and Equations

Factoring Trinomials ax² + bx + c

A1.3About 70 minutes
Learning goals
  • I can factor ax² + bx + c by decomposition.
  • I can use guess and check or a box to factor when a ≠ 1.
  • I can factor fully when there is a common factor or a negative leading coefficient.

Words to know

decomposition
Factoring ax² + bx + c by splitting bx into two terms whose coefficients multiply to ac, then grouping.
grouping
Factoring four terms by taking a common factor from each pair, then a common bracket.
leading coefficient
The number a in ax² + bx + c.
factor over the integers
Write as a product of factors whose coefficients are all integers.

When a is not 1

The product-and-sum trick from the last lesson works because the x² term is just x². When the leading coefficient is another number, as in 3x² + 13x − 10, the factors look like (3x ?)(x ?) or (px + q)(rx + s), and there are more possibilities to sort through. This lesson gives you a method that is guaranteed to work if the trinomial factors at all: decomposition (also called splitting the middle term). It also shows a quicker guess-and-check approach and a visual box method.

Decomposition, step by step

To factor ax² + bx + c (after removing any common factor):

  1. Multiply a × c.
  2. Find two integers whose product is ac and whose sum is b.
  3. Split the middle term bx into two terms using those integers.
  4. Group the four terms in pairs and take out a common factor from each pair.
  5. Take out the common bracket.

Why does this work? Splitting the middle term does not change the value of the expression, and choosing numbers with product ac guarantees that the two pairs share a bracket.

Decomposition: 3x² + 13x − 101. Multiply a × c3 × (−10) = −302. Find two numbersproduct −30, sum 13: 15 and −23. Split the middle term3x² + 15x − 2x − 104. Group in pairs3x(x + 5) − 2(x + 5)5. Common bracket(x + 5)(3x − 2)
Decomposition of 3x² + 13x − 10. This is the example from the curriculum, and the same five steps work on every factorable trinomial.
Worked example

Example 1: Decomposition

Factor 6x² − 11x + 4.

  1. No common factor. a × c = 6 × 4 = 24.
  2. Product 24 and sum −11: both numbers are negative. Pairs: −1, −24 (−25); −2, −12 (−14); −3, −8 (−11) ✓.
  3. Split: 6x² − 3x − 8x + 4.
  4. Group: (6x² − 3x) + (−8x + 4) = 3x(2x − 1) − 4(2x − 1). Take out −4 from the second pair so that the brackets match.
  5. Common bracket: (2x − 1)(3x − 4).
  6. Check: (2x − 1)(3x − 4) = 6x² − 8x − 3x + 4 = 6x² − 11x + 4 ✓.

(2x − 1)(3x − 4)

3x² + 13x − 10 in a boxx+53x3x²15x−2−2x−10Cells: 3x², 15x, −2x, −10Common factor of each rowand column gives the sides:(3x − 2)(x + 5)
The box method organizes the same four terms. Put ax² top-left and c bottom-right, put the split middle terms in the other two cells, then take the common factor of each row and column.
Worked example

Example 2: Factor fully, with a common factor

Factor fully: 12x² + 2x − 4.

  1. Common factor 2: 12x² + 2x − 4 = 2(6x² + x − 2).
  2. For 6x² + x − 2: a × c = −12, sum 1: 4 and −3.
  3. Split and group: 6x² + 4x − 3x − 2 = 2x(3x + 2) − 1(3x + 2) = (3x + 2)(2x − 1).
  4. Do not forget the 2 in front.

2(3x + 2)(2x − 1)

Worked example

Example 3: Negative leading coefficient

Factor −4x² + 4x + 15.

  1. Take out −1: −(4x² − 4x − 15).
  2. a × c = −60, sum −4: −10 and 6.
  3. 4x² − 10x + 6x − 15 = 2x(2x − 5) + 3(2x − 5) = (2x − 5)(2x + 3).
  4. Answer: −(2x − 5)(2x + 3). Check with x = 0: original 15; answer −(−5)(3) = 15 ✓.

−(2x − 5)(2x + 3)

Guess and check (for confident factorers)

When a and c have few factors, you can test bracket arrangements directly. For 2x² + 7x + 3, the brackets must be (2x ?)(x ?) and the constants must multiply to 3, so try 1 and 3:

  • (2x + 1)(x + 3): middle term 6x + x = 7x ✓
  • (2x + 3)(x + 1): middle term 2x + 3x = 5x ✗

Only the middle term needs checking, because the first and last terms are right by construction. If the numbers are big (such as 12x² − 8x − 15), decomposition is usually faster than guessing.

Decomposition tips

  • Take out the GCF first: it makes ac smaller.
  • If ac is positive, both numbers have the sign of b. If ac is negative, the signs differ.
  • When you group, if the brackets don't match, check the sign you took out of the second pair.
  • If no pair of integers works, the trinomial does not factor over the integers.
Just for fun

Decomposition is the only recipe where you split something in two, regroup it, and somehow end up with fewer pieces than you started with.

Hands-on activity

Build it with paper tiles

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

  1. Cut 2 large squares 6 cm by 6 cm (each is x²), 7 strips 6 cm by 1 cm (each is x) and 3 small squares 1 cm by 1 cm (each is 1).
  2. Arrange all 12 pieces into one rectangle with no gaps or overlaps. (Hint: put the two big squares side by side.)
  3. Measure the rectangle's sides in terms of x: one side is 2x + 1 and the other is x + 3.
  4. Write 2x² + 7x + 3 = (2x + 1)(x + 3) and check by expanding.

What to look for: the two x² squares sit side by side to make a 12 cm by 6 cm block. One x-strip stands upright beside them (making the long side 2x + 1). The other 6 strips lie flat below the squares in 3 rows of 2, and the 3 unit squares fill the corner under the upright strip (making the short side x + 3). A rectangle that uses every piece means the trinomial factors; if no rectangle is possible, it does not.

Your turn

1

Factor 2x² + 7x + 3.

2

Factor 3x² − 10x + 8.

3

Factor 5x² + 13x − 6.

4

Factor 4x² − 4x − 3.

5

Factor 6x² + 7x + 2.

6

Factor fully: 10x² − 25x − 15.

7

Factor −3x² + 5x + 2.

More practice: warm-up

1

Factor 2x² + 5x + 2.

2

Factor 3x² + 4x + 1.

3

Factor 2x² − 7x + 3.

4

Factor 5x² + 4x − 1.

5

For 2x² + 11x + 12, which two numbers split the middle term? Then factor.

More practice: on your own

1

Factor 8x² − 10x − 3.

2

Factor 9x² + 9x − 10.

3

Factor 6x² − 19x + 15.

4

Factor fully: 18x² + 3x − 6.

5

Factor 12x² − 8x − 15.

Challenge

1

Factor fully: 100x² − 115x − 35.

2

Find all integers k so that 2x² + kx + 3 factors over the integers.

3

Factor 2(x + 1)² + 5(x + 1) − 3.

Check your understanding

Quiz

1.

Factor 2x² + 5x − 3.

2.

To factor 6x² + 5x − 6 by decomposition, which two numbers split the middle term?

3.

What is the first step when factoring 8x² + 20x + 8?

4.

Which product equals 4x² + 8x + 3?

5.

In decomposition, if no pair of integers has product ac and sum b, then:

6.

For guess and check on (3x + p)(x + q), which term do you need to check?

For parents and teachers

Big idea (A1.3): trinomials ax² + bx + c with a ≠ 1, such as the curriculum's example 3x² + 13x − 10, factored by decomposition, guess and check, or the box method.

How to help: let your teen choose the method they prefer, but ask them to show the split (the two numbers) so errors are easy to find. Expanding the answer is always the final check.

Common difficulties:

  • Using the sum and product of c alone (the x² + bx + c method) when a ≠ 1. The product must be a × c.
  • Sign errors when grouping, especially taking a negative out of the second pair.
  • Leaving out a common factor that was removed at the start.
  • Spending ages on guess and check with large numbers. Decomposition is systematic.