Algebra

Factorising

Find the product inside an expression.

Predict, test and explain

Step 1 of 3

Predict

For (x + 2)(x − 3) = 0, which pair of x-values solves the equation?

x2−x−6x^2 - x - 6
−6-6
−5-5
−4-4
−3-3
−2-2
−1-1
00
11
22
33
44
55
66
xx
yy
roots: product=0\text{roots: product}=0

Scroll the diagram sideways to see every label.

p+q=−1p+q = -1pq=−6pq = -6

InvestigateWhich two numbers multiply to the constant and add to the middle coefficient?

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

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Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

Factorise this quadratic.

x2−5x+6x^2-5x+6

Hint

Find two numbers whose product is six and whose sum is negative five.

Reveal answer

Negative two and negative three have the required product and sum.

x2−5x+6=(x−2)(x−3)x^2-5x+6=(x-2)(x-3)

Build an explanation

Hint 1

Compare the two bracket numbers with the middle and constant terms.

Hint 2

Their sum gives the coefficient of x; their product gives the constant.

Hint 3

For x² + bx + c, find p and q with p + q = b and pq = c.

Worked example

This example uses fixed values, separate from the diagram controls.

Factorise x² − x − 6.

  1. Find two numbers multiplying to −6 and adding to −1.
  2. 2 and −3 satisfy both conditions.
  3. Use them in the two brackets.

Answer: (x + 2)(x − 3).

Connect this idea

Watch out: A matching product alone is insufficient: the sum must match the x-coefficient too.