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The Class 10 NCERT chapter on quadratic equations says that Sridharacharya "derived a formula, now known as the quadratic formula, (as quoted by Bhaskara II)". Most students learn the formula and never see the rule itself. It is worth seeing, because it is a cleaner way to complete the square than the one many of us were taught, and it explains where the 4a and the b² in the formula come from. This post gives the rule in an old English translation and works it on two textbook problems.
Who was Sridharacharya?
Very little is known about his life, and even his century is disputed. The NCERT Class 10 chapter (2026-27 reprint) gives "C.E. 1025". The MacTutor biography says dates from the seventh to the eleventh century have been proposed and that the best present estimate is that he wrote around 900 CE. Two works survive, the Patiganita and the Trisatika, on arithmetic and its uses: fractions, interest, mixtures, series and areas. His algebra, in which the quadratic rule appeared, is lost. We know the rule because Bhaskara II quoted it around 1150.
What exactly did the rule say?
In H. T. Colebrooke's 1817 translation of Bhaskara II's algebra, the Bija-ganita, section 131 reads:
"Sridhara's rule on this point: 'Multiply both sides of the equation by a number equal to four times the [coefficient] of the square, and add to them a number equal to the square of the original [coefficient] of the unknown quantity.' (Then extract the root.)"
The equation is written with the unknown terms on one side: ax² + bx = c. Follow the rule:
Multiply by 4a: 4a²x² + 4abx = 4ac.
Add b² to both sides: 4a²x² + 4abx + b² = b² + 4ac.
The left side is now a perfect square: (2ax + b)² = b² + 4ac.
Take the root: 2ax + b = ±√(b² + 4ac), so x = [−b ± √(b² + 4ac)] / 2a.
Write the equation as ax² + bx + c = 0 instead, so the constant changes sign, and you get the familiar x = [−b ± √(b² − 4ac)] / 2a.
Notice the word "original" in the rule. A commentator in Colebrooke's notes stresses that you add the square of the coefficient of x as it was before multiplying, that is b², not (4ab)².
Worked example 1: NCERT's prayer hall
The NCERT chapter opens with a hall whose breadth x metres satisfies 2x² + x − 300 = 0.
Rewrite as 2x² + x = 300. Here a = 2, b = 1, c = 300.
Multiply by 4a = 8: 16x² + 8x = 2400.
Add b² = 1: 16x² + 8x + 1 = 2401.
So (4x + 1)² = 2401 = 49².
4x + 1 = 49 gives x = 12; 4x + 1 = −49 gives x = −12.5.
A breadth cannot be negative, so the breadth is 12 m. Check: 2(144) + 12 = 300.
Multiplying by 4a keeps every number whole. Dividing by a first, the usual school method, would have given x² + x/2 = 150 and fractions at every step.
Worked example 2: Bhaskara's monkeys
Bhaskara II follows the rule with examples. In section 140 of the same translation: the fifth part of a troop of monkeys, less three, squared, had gone into a cave, and one monkey was in sight on a branch. How many were there?
Let the troop be 5y. Then (y − 3)² + 1 = 5y, so y² − 11y = −10.
Multiply by 4: 4y² − 44y = −40. Add 11² = 121: 4y² − 44y + 121 = 81.
(2y − 11)² = 81, so 2y − 11 = ±9, giving y = 10 or y = 1.
The troop is 50 or 5.
Bhaskara keeps 50 and rejects 5, saying the second value is "not to be taken". With 5 monkeys, a fifth less three is 1 − 3 = −2, and a negative number of monkeys makes no sense, even though the squared equation is satisfied.
Which mistakes does this help you avoid?
Dropping the minus root. MacTutor notes there is no suggestion that Sridhara himself took two values. You must: in section 139, Bhaskara's earlier monkey problem gives 48 and 16, and both work.
Adding b² to one side only. The rule says add it to both sides.
Squaring the wrong coefficient. Add b², not the new coefficient 4ab.
Sign of c. Move the constant before you start, and keep its sign.
For more practice, our Class 10 quadratic equations guide covers factorisation and the discriminant, and our post on Bhaskara II and the Lilavati covers the writer who preserved this rule.
What can a student learn from Sridharacharya?
His rule is a method, not a formula. If you remember "multiply by 4a, add b², take the root", you can rebuild the quadratic formula in an exam even if you forget it, and you can also solve equations that are not yet in standard form.
When you know the formula but cannot see why a step works, a worked explanation helps more than another answer key. The Fermi Problem Library has a step-by-step solver that explains the reasoning behind each line.
Try it now: solve 3x² + 4x = 15 with Sridharacharya's rule, keeping every number whole.
