CBSE Class 10 Mathematics Quadratic Equations Notes
Quadratic Equations is Chapter 4 of Class 10 Mathematics. A quadratic equation is a polynomial equation of degree 2. This chapter covers the standard form of a quadratic equation, three methods to find its roots (zeros) - factorisation, completing the square, and the quadratic formula, and the discriminant which determines the nature of roots.
Key Topics Covered
• Standard Form of a Quadratic Equation
• Solution by Factorisation Method
• Solution by Completing the Square
• Quadratic Formula (Sridharacharya's Formula)
• Discriminant and Nature of Roots
• Quick Formula Summary & Board Exam Practice
1. Standard Form of a Quadratic Equation
A quadratic equation in the variable x is an equation of the form:
ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0
Here, a is the coefficient of x², b is the coefficient of x, and c is the constant term. The condition a ≠ 0 ensures the equation is truly quadratic (degree 2).
The values of x that satisfy ax² + bx + c = 0 are called the roots or zeros of the quadratic equation.
Identifying Quadratic Equations
Check whether an equation is quadratic by verifying it can be written in the standard form ax² + bx + c = 0 with a ≠ 0:
Equation | Standard Form | Quadratic? |
x² – 5x + 6 = 0 | x² – 5x + 6 = 0 | Yes (a=1, b=−5, c=6) |
2x(x – 3) = 0 | 2x² – 6x = 0 | Yes (a=2, b=−6, c=0) |
x² + 1/x = 0 | Not a polynomial | No |
(x+1)(x−1) = x² | 0·x² – 0x – 1 = 0 | No (a=0) |
2. Solution by Factorisation Method
In the Factorisation Method, we express the quadratic equation as a product of two linear factors and then use the Zero Product Property: if A × B = 0, then A = 0 or B = 0.
Steps for Factorisation
1. Write the equation in standard form: ax² + bx + c = 0.
2. Find two numbers p and q such that p + q = b and p × q = ac.
3. Split the middle term bx as px + qx and factorise.
4. Set each factor to zero and solve for x.
Example: Solve x² – 5x + 6 = 0 by factorisation.
Find p, q: p + q = −5, p × q = 6 → p = −2, q = −3
Split: x² – 2x – 3x + 6 = 0
Factorise: x(x – 2) – 3(x – 2) = 0 → (x – 2)(x – 3) = 0
Roots: x = 2 or x = 3 ✓
3. Solution by Completing the Square
Completing the Square is a method used to convert the quadratic into the form (x + k)² = d, from which roots are found by taking the square root of both sides.
Steps for Completing the Square
1. Write the equation in standard form ax² + bx + c = 0.
2. Divide throughout by a (coefficient of x²) to make the leading coefficient 1.
3. Move the constant term to the right-hand side.
4. Add (b/2a)² to both sides to complete the square on the left.
5. Write the left side as a perfect square trinomial and simplify the right.
6. Take the square root of both sides and solve for x.
Example: Solve 2x² + 8x + 6 = 0 by completing the square.
Divide by 2: x² + 4x + 3 = 0
Move constant: x² + 4x = −3
Add (4/2)² = 4 to both sides: x² + 4x + 4 = −3 + 4 = 1
Perfect square: (x + 2)² = 1
Take root: x + 2 = ±1 → x = −1 or x = −3 ✓
4. Quadratic Formula (Sridharacharya's Formula)
The Quadratic Formula gives the roots of any quadratic equation ax² + bx + c = 0 directly, without factorising. It is derived by applying completing the square to the general form:
x = (–b ± √(b² – 4ac)) / 2a
This formula always works, even when factorisation is difficult or impossible.
Steps to Apply the Quadratic Formula
1. Identify a, b, c from the standard form ax² + bx + c = 0.
2. Calculate the discriminant: D = b² – 4ac.
3. If D ≥ 0, substitute into the formula to find two roots.
4. If D < 0, the equation has no real roots.
Example: Solve 2x² – 7x + 3 = 0 using the quadratic formula.
a = 2, b = −7, c = 3
D = b² – 4ac = 49 – 24 = 25
x = (7 ± √25) / 4 = (7 ± 5) / 4
x₁ = (7 + 5)/4 = 3 and x₂ = (7 – 5)/4 = 1/2 ✓
5. Discriminant and Nature of Roots
The Discriminant, denoted by D or Δ (Delta), determines the nature of the roots without actually solving the equation:
D = b² – 4ac
Nature of Roots Based on Discriminant
Discriminant | Nature of Roots | Type | Example |
D > 0 | Two distinct real roots | Unequal | x²−5x+6=0, D=1 |
D = 0 | Two equal real roots | Repeated | x²−4x+4=0, D=0 |
D < 0 | No real roots | Imaginary | x²+x+1=0, D=−3 |
Equal roots occur when D = 0, i.e., b² = 4ac. The repeated root is x = –b / 2a.
Relationship Between Roots and Coefficients
If α and β are the roots of ax² + bx + c = 0, then:
Sum of roots: α + β = –b/a
Product of roots: αβ = c/a
Quadratic with roots α, β: x² – (α+β)x + αβ = 0
6. Quick Formula Summary
All key formulas from this chapter that are frequently tested in CBSE board examinations:
Standard Form | ax² + bx + c = 0, a ≠ 0 |
Quadratic Formula | x = (–b ± √(b²−4ac)) / 2a |
Discriminant | D = b² – 4ac |
D > 0 → | Two distinct real roots |
D = 0 → | Two equal (repeated) real roots |
D < 0 → | No real roots |
Sum of roots (α+β) | –b / a |
Product of roots (αβ) | c / a |
Form QE from roots | x² – (Sum)x + (Product) = 0 |
Equal roots condition | b² = 4ac, repeated root = –b/2a |
7. Board Exam Practice Questions
Commonly asked question types in CBSE Class 10 board examinations from this chapter:
1 Mark Questions
1. For what value of k does kx² + 2x + 1 = 0 have equal roots?
2. Find the discriminant of the equation 3x² – 2x + 1 = 0.
3. State the nature of roots of x² + 4x + 5 = 0.
4. If 2 is a root of x² + kx – 6 = 0, find the value of k.
3 Mark Questions
1. Solve 2x² – 5x + 3 = 0 by the factorisation method.
2. Solve x² – 4x – 8 = 0 by completing the square method.
3. Find the value of k for which the equation 2x² + kx + 3 = 0 has two equal roots.
4. The sum of a number and its reciprocal is 10/3. Find the number.
5 Mark Questions
1. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less. Find the speed of the train.
2. Solve using the quadratic formula: 3x² – 5x + 2 = 0. Also state the nature of its roots.
3. The product of two consecutive positive integers is 306. Find the integers using the quadratic equation method.
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