Factorization by Splitting the Middle Term

Factorization is the fastest way to solve the quadratics that bank exams set, and it beats the quadratic formula on every count for speed. The method is simple: split the middle coefficient into two parts whose sum is the middle term and whose product equals the constant term. With a little practice you will read the split off the numbers in a couple of seconds.

The Core Rule

For x² + bx + c = 0, find two numbers whose sum is b and whose product is c. Those two numbers, with signs attached, immediately give the roots. Because a = 1 in almost all bank quadratics, you never have to worry about distributing a leading coefficient, which is precisely what makes the split so quick here compared with school algebra.

x² + bx + c = 0 → find p, q with p + q = b and p × q = c → roots are −p and −q

Getting the Signs Right

The signs of b and c tell you the signs of the two numbers before you even search for them. If c is positive, both numbers share the sign of b; if c is negative, the numbers have opposite signs and the larger one carries the sign of b. Fixing signs first cuts your search in half and removes the most common source of error in the whole topic.

Working Backwards From Factors

Since the roots satisfy (x − root1)(x − root2) = 0, once you have the two numbers you can write the roots directly by flipping their signs. A split of +3 and +4 in x² + 7x + 12 gives factors (x + 3)(x + 4), so the roots are −3 and −4. Practise reading straight from split to roots so the sign flip becomes automatic.

Solved Examples

Example 1: x² + 9x + 20 = 0. Numbers 4 and 5 (sum 9, product 20), so roots are −4 and −5.

Example 2: x² − 2x − 15 = 0. Product −15, sum −2 → numbers −5 and +3, so roots are 5 and −3.

✗ x² − 2x − 15: guessing roots 5 and 3 (both positive)  |  ✓ Opposite signs: roots 5 and −3, since product is negative

A Worked Split, Step by Step

Take x² − 11x + 28 = 0 and walk the routine slowly once so it becomes fast forever. The constant 28 is positive and the middle term is negative, so both numbers are negative. List factor pairs of 28: 1 and 28, 2 and 14, 4 and 7. The pair that adds to 11 is 4 and 7, so with negative signs the split is −4 and −7. Flipping the signs of the split gives the roots 4 and 7. Every clean bank quadratic yields to exactly this sequence — list factor pairs of the constant, pick the pair that matches the middle term, attach the correct signs, and flip. Rehearse it on ten equations and the whole process collapses to a few seconds.