# CBSE Class 10t Chapter 4 Quadratic Equations Details & Preparations Downloads

As we step into the realm of quadratic equations, the intricacies of this mathematical chapter unfold. In this blog post, we'll delve into the definition, properties, and various methods of solving quadratic equations, unraveling the beauty and importance of this fundamental concept in mathematics.

**Quadratic Equations Demystified Navigating the Complexity of a Fundamental Chapter**

**Defining Quadratic Equations**

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where 'a' is called the leading coefficient and 'c' is called the absolute term of f (x).

**Understanding the Quadratic Formula**

The quadratic formula helps us solve any quadratic equation. First, we bring the equation to the form ax²+bx+c=0, where a, b, and c are coefficients. Then, we plug these coefficients in the formula: (-b±√(b²-4ac))/(2a) . See examples of using the formula to solve a variety of equations.

**Quadratic equations**

Quadratic equations are the polynomial equations of degree 2 in one variable of type f(x) = ax2 + bx + c = 0 where a, b, c, ∈ R and a ≠ 0. It is the general form of a quadratic equation where ‘a’ is called the leading coefficient and ‘c’ is called the absolute term of f (x). The values of x satisfying the quadratic equation are the roots of the quadratic equation (α, β).

**Graphical Representation**

Visualizing quadratic equations on a coordinate plane provides a geometric understanding of parabolas, the graphical representation of quadratic functions. This exploration allows us to interpret key features such as vertex, axis of symmetry, and direction of opening.

**Quadratic Equation Definition**

A quadratic polynomial, when equated to zero, becomes a quadratic equation. In other terms, a quadratic equation is a second degree algebraic equation. The values of x satisfying the equation are called the roots of the quadratic equation.

**General from:** ax^{2} + bx + c = 0

Here, a, b, c, ∈ R and a ≠ 0

**Examples:** 3x^{2} + x + 5 = 0, -x^{2} + 7x + 5 = 0, x^{2} + x = 0.

**Quadratic Equation Formula**

The solution or roots of a quadratic equation are given by the quadratic formula:

x = (α, β) = [-b ± √(b^{2} – 4ac)]/2a

**Roots of Quadratic Equation**

The values of variables satisfying the given quadratic equation are called their roots. In other words, x = α is a root of the quadratic equation f(x), if f(α) = 0.

The real roots of equation f(x) = 0 are the x-coordinates of the points where the curve y = f(x) intersects the x-axis.

- One of the roots of the quadratic equation is zero, and the other is -b/a if c = 0
- Both the roots are zero if b = c = 0
- The roots are reciprocal to each other if a = c

**CBSE Class 10 NCERT Mathematics Topics for a Strong Foundation (NCERT DOWNLOAD)**

Chapter Name |
Quadratic Equations |

Topic Number |
Topics |

4.1 |
Introduction |

4.2 |
Quadratic Equations |

4.3 |
Solution of a Quadratic Equation by Factorisation |

4.4 |
Nature of Roots |

4.5 |
Summary |

**Quadratic Equations Having Common Roots**

Let β be the common root (solution) of quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0. This implies that a1β2 + b1β + c1 = 0 and a2β2 + b2β + c2 = 0.

Now, solving for β2 and β we will get:

β2/(b1c2 – b2c1) = -β/(a1c2 – a2c1) = 1/(a1b2 – a2b1) [using determinant method]

Therefore, β2 = (b1c2 – b2c1)/ (a1b2 – a2b1) . . . . . . . . . . . . . . . . (1)

And, β = (a2c1 – a1c2)/(a1b2 – a2b1) . . . . . . . . . . . . . . . . (2)

On squaring equation (2) and equating it with equation (1), we get:

(a1b2 – a2b1)/(b1c2 – b2c1) = (a2c1 – a1c2)2

Hence, it is the required condition for quadratic equations having one common root.

If both the roots of quadratic equations a1x2 + b1x + c1 = 0 and a2x2 + b2x + c2 = 0 are common then:

a1/a2 = b1/b2 = c1/c2

If α is a repeated root, i.e., the two roots are α, α of equation f(x) = 0, then α will be a root of the derived equation.

f’(x) = 0 where f’(x) = df/dx

If α is a repeated root common in f(x) = 0 and ϕ(x) = 0, then α is a common root both in f’(x) = 0 and ϕ ‘(x) = 0.

**CBSE Class 10 Board Exam Sample Paper**

**[Previous Year Question Solution Maths Download Button]
[Previous Year Question Solution Science Download Button]**

CBSE CLASS 10 Mathematics Chapters |

Chapter1: Real Numbers |

Chapter2: Polynomials |

Chapter3: Pair of Linear Equations in Two Variables |

Chapter4: Quadratic Equations |

Chapter5: Arithmetic Progressions |

Chapter6: Triangles |

Chapter7: Coordinate Geometry |

Chapter8: Introduction to Trigonometry |

Chapter9: Some Applications of Trigonometry |

Chapter10: Circles |

Chapter11: Areas Related to Circles |

Chapter12: Surface Areas and Volumes |

Chapter13: Statistics |

Chapter14: Probability |

CBSE CLASS 10 Science Chapters |

Chapter1: Chemical Reactions and Equations |

Chapter2: Acids, Bases and Salts |

Chapter3: Metals and Non-metals |

Chapter4: Carbon and its Compounds |

Chapter5: Life Processes |

Chapter6: Control and Coordination |

Chapter7: How do Organisms Reproduce? |

Chapter8: Heredity |

Chapter9: Light – Reflection and Refraction |

Chapter10: The Human Eye and the Colourful World |

Chapter11: Electricity |

Chapter12: Magnetic Effects of Electric Current |

Chapter13: Our Environment |

Class 8 |

Class 9 |

Class 11 |

Class 12 |

**CBSE Class 10th Downloadable Resources: **

1. CBSE Class 10th Topic Wise Summary | View Page / Download |

2. CBSE Class 10th NCERT Books | View Page / Download |

3. CBSE Class 10th NCERT Solutions | View Page / Download |

4. CBSE Class 10th Exemplar | View Page / Download |

5. CBSE Class 10th Previous Year Papers | View Page / Download |

6. CBSE Class 10th Sample Papers | View Page / Download |

7. CBSE Class 10th Question Bank | View Page / Download |

8. CBSE Class 10th Topic Wise Revision Notes | View Page / Download |

9. CBSE Class 10th Last Minutes Preparation Resources (LMP) | View Page / Download |

10. CBSE Class 10th Best Reference Books | View Page / Download |

11. CBSE Class 10th Formula Booklet | View Page / Download |

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**FAQ**

**Q1. What are the roots of a quadratic equation?**

**Ans ** The roots are the values of the variable that make the quadratic equation equal to zero.

**Q2. How can I find the roots of a quadratic equation?**

**Ans** The roots can be found using the quadratic formula or by factoring the equation.

**Q3. Can a quadratic equation have only one root?**

**Ans** Yes, if the discriminant (the part under the square root in the quadratic formula) is zero, the equation has a single repeated root.

**Q4. What is the discriminant, and how does it determine the nature of roots?**

**Ans **The discriminant is the expression inside the square root in the quadratic formula It determines whether the roots are real, equal, or imaginary.

**Q5. How does factoring help find the roots of a quadratic equation?**

**Ans **Factoring involves expressing the quadratic equation as a product of two linear factors, making it easier to identify the values that satisfy the equation.