Splitting the Middle Term
To factorise ax² + bx + c = 0, we find two numbers p and q such that:
If (x - α)(x - β) = 0, then either x - α = 0 or x - β = 0. Hence, x = α or x = β are the roots of the equation.
Solved Practice Examples
Find the roots of x² - 5x + 6 = 0
Step 1: Identify coefficients: a = 1, b = -5, c = 6. We need p × q = 6 and p + q = -5.
The numbers are -2 and -3.
Step 2: Split the middle term:
x(x - 2) - 3(x - 2) = 0
(x - 2)(x - 3) = 0
Step 3: Equate factors to 0:
x - 3 = 0 ⟹ x = 3
Roots are x = 2 and x = 3.
Find the roots of 2x² - x - 6 = 0
Step 1: Identify coefficients: a = 2, b = -1, c = -6. Product = a × c = 2 × (-6) = -12. We need p + q = -1.
The numbers are -4 and 3.
Step 2: Split the middle term & factorise:
2x(x - 2) + 3(x - 2) = 0
(2x + 3)(x - 2) = 0
Step 3: Solve for x:
x - 2 = 0 ⟹ x = 2
Key Steps
Find two numbers whose product is a × c and sum is b.
Rewrite the middle term bx using these two numbers and factorise by grouping.
Set each linear factor to zero to find the roots x = α and x = β.