Split, factor, set each to zero
Factorise the left side; each factor gives a root.
When is a product zero?
If two numbers multiply to give 0, such as p × q = 0, what can we say about p and q?
Pick the one that is always true.
What this lesson covers
The idea
If ax² + bx + c can be factorised into two linear factors (by splitting the middle term), the roots of ax² + bx + c = 0 are found by equating each factor to zero.
When is a product zero?
If two numbers multiply to give 0, such as p × q = 0, what can we say about p and q?
Pick the one that is always true.
- At least one of them is 0
- Both of them are 0
- Neither of them is 0
Split the middle term
Pick the two numbers that multiply to a × c and add to b. Then follow the working: split, take out common factors, make a product, and set each factor to 0.
Factors give roots
If ax² + bx + c can be written as two linear factors (by splitting the middle term), the roots of ax² + bx + c = 0 are found by setting each factor equal to 0.
Why does this work? (2x − 3)(x − 1) = 0 is a product equal to 0, so 2x − 3 = 0 or x − 1 = 0. That gives x = 3/2 or x = 1.
To split the middle term, find two numbers with product a × c and sum b. For 2x² − 5x + 3: a × c = 6 and b = −5, so −2 and −3.
Notes
Split the middle term (two numbers with product ac and sum b), make two linear factors, and set each factor equal to 0. Each gives a root.
Check yourself
Split the middle term of 3x² + 10x + 8. The two numbers must multiply to 24 and add to 10. One is 4. What is the other?
Answer: 6
4 × 6 = 24 and 4 + 6 = 10. So 3x² + 10x + 8 = 3x² + 4x + 6x + 8 = x(3x + 4) + 2(3x + 4) = (x + 2)(3x + 4).
Solve x² − x − 12 = 0 by splitting −x as −4x + 3x. What is the positive root?
Answer: 4
(x + 3)(x − 4) = 0, so x = −3 or x = 4. The positive root is 4.
The hall: breadth x metres with 2x² + x − 300 = (x − 12)(2x + 25) = 0. The roots are 12 and −12.5. What is the breadth, in metres?
Answer: 12 m
x = 12 or x = −12.5. The breadth cannot be negative, so it is 12 m. The length is 2 × 12 + 1 = 25 m, and 12 × 25 = 300 m².
x² − 6x + 9 = (x − 3)(x − 3). Setting each factor to 0, the roots are…
- x = 3 and x = 3: the same root twice — correct. Yes! Each factor x − 3 gives x = 3.
- x = 3 and x = −3. x − 3 = 0 gives x = 3 for both factors. Neither gives −3.
- x = 9. 9 is the plain number c. Set each factor x − 3 to 0.