Class 10 Maths Chapter 4 Quadratic Equations Formula Sheet

The Class 10 Maths Chapter 4 Quadratic Equations formula sheet puts every key result on one page. It covers the standard form ax² + bx + c = 0, what a root means, the three solving methods (factorisation, completing the square and the quadratic formula), and the discriminant test for the nature of roots. Built for the 2026-27 CBSE syllabus, it is made for night-before revision.

• All core formulas in one place, with a plain-English meaning for each.
• Discriminant test that tells you two distinct, two equal or no real roots from b² − 4ac.
• Method toolkit: factorisation, completing the square and the quadratic formula, with one example each.

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Complete Formula List

The table below lists every named result you need from Quadratic Equations. The chapter rests on two ideas: any quadratic can be written as ax² + bx + c = 0 and solved three ways (factorisation, completing the square, the quadratic formula), and the discriminant b² − 4ac tells you how many real roots there are before you solve.

Standard Form, Roots and Zeroes

A quadratic equation is any equation in the standard form ax² + bx + c = 0, where a, b, c are real and a is not zero (if a were 0 the equation would be linear). A root is a value of x that makes the left side zero, the same as the zeroes of the polynomial. A few facts to keep straight:

• Degree 2 means at most two roots: two distinct, two equal, or no real root.
• Read coefficients with signs: for 3x² − 4x + 1 = 0, a = 3, b = −4, c = 1, not b = 4.
• Rearrange first: x(x + 2) = 5 becomes x² + 2x − 5 = 0.

Always write the equation as "= 0" and note a, b, c carefully. A sign slip in b or c breaks both the quadratic formula and the discriminant test.

Solving by Factorisation

Factorisation is the fastest method when the numbers are friendly. Write the quadratic as a product of two linear factors, then use the zero-product rule: if a product is zero, at least one factor is zero. The trick is splitting the middle term bx into two parts.

• Step 1: find two numbers whose product is a × c and sum is b.
• Step 2: split bx using those numbers, then group and take out common factors.
• Step 3: write (px + q)(rx + s) = 0 and set each factor to zero.

Example: for 2x² − 5x + 3 = 0, ac = 6 and the sum is −5, so split −5x = −2x − 3x. Then 2x(x − 1) − 3(x − 1) = (x − 1)(2x − 3) = 0, giving x = 1 or x = 3/2. Never divide out a common x: x² = 5x must become x(x − 5) = 0 to keep the root x = 0.

Completing the Square Method

Completing the square is the idea behind the quadratic formula. Rewrite the equation so one side is a perfect square equal to a number, then take the square root. Use it when the quadratic does not factor neatly.

For ax² + bx + c = 0, completing the square gives (x + b/2a)² = (b² − 4ac) / 4a². The steps:

• Use the identity (x ± m)² = x² ± 2mx + m².
• Half then square: to complete the square on x² + px, add (p/2)².
• Square root step: once you have "square = number", take the square root of both sides (with a ±).

Because it always works, completing the square is also how the quadratic formula is derived. If you forget the formula, rebuild it this way in a couple of steps.

The Quadratic Formula and the Two Roots

The quadratic formula gives both roots straight from a, b and c. For ax² + bx + c = 0 with a ≠ 0 and b² − 4ac ≥ 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. It is also called the Sridharacharya formula.

Example: for x² − 3x − 10 = 0, a = 1, b = −3, c = −10, so b² − 4ac = 49 and √49 = 7. The roots are x = (3 ± 7) / 2, giving x = 5 or x = −2. Never drop the ±, or you get just one answer.

Sum and Product of Roots

You can read facts about the roots without solving. If α and β are the roots of ax² + bx + c = 0, then the sum α + β = −b/a and the product αβ = c/a, a reliable way to check an answer.

• Verify your roots: add them and compare with −b/a, multiply them and compare with c/a. A mismatch means an arithmetic slip.
• Build the equation: from a known sum and product, the quadratic is x² − (α + β)x + αβ = 0.
• Word problems: these formulas turn a relation between roots into an equation in a, b and c.

For x² − 3x − 10 = 0 with roots 5 and −2, the sum 3 matches −b/a, and the product −10 matches c/a. This 30-second check catches most mistakes.

Discriminant and Nature of Roots

The discriminant is the quantity under the square root: D = b² − 4ac. Its sign decides the nature of the roots without solving, in three cases.

Example: for 2x² − 4x + 3 = 0, D = 16 − 24 = −8, so the equation has no real roots. For a "find k" question, set D = 0 for equal roots, D > 0 for two distinct real roots, D < 0 for none.

How to Use This Formula Sheet

• Night-before revision: read every row and check you can state each rule, especially the quadratic formula and the three discriminant cases, without your notes.
• During practice: keep the PDF open in a second tab to look up the formula and the sum-product relations while you solve.
• For "find k" questions: equal roots means D = 0, distinct real roots means D > 0, and no real roots means D < 0.

Board Exam Weightage

Quadratic Equations sits in the Algebra unit. The table shows the high-frequency question types, so you can plan revision time.

This chapter usually carries about 5 to 7 marks, often through a word problem framed as a quadratic. The discriminant test and the quadratic formula are reliable scoring chances.

Common Mistakes to Avoid

More Quadratic Equations Resources

Use this formula sheet with the other Quadratic Equations resources below.

Formula Sheets for Class 10 Maths: All Chapters