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Q1(i):
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) $2x^2 – 3x + 5 = 0$

Solution :

Given: A quadratic equation $2x^2 - 3x + 5 = 0$.

To Find: The nature of the roots of the given quadratic equation and, if real roots exist, to determine their values.

Step 1: Identify the coefficients of the quadratic equation.
A standard quadratic equation is represented as $ax^2 + bx + c = 0$.
Comparing the given equation $2x^2 - 3x + 5 = 0$ with the standard form:
$a = 2$
$b = -3$
$c = 5$

Step 2: Determine the nature of the roots using the Discriminant ($D$).
The discriminant of a quadratic equation is given by the formula:
$D = b^2 - 4ac$
Substituting the values of $a$, $b$, and $c$ into the formula:
$D = (-3)^2 - 4(2)(5)$
$D = 9 - 40$
$D = -31$

Step 3: Analyze the nature of the roots based on the value of $D$.
According to the theory of quadratic equations:
1. If $D > 0$, the equation has two distinct real roots.
2. If $D = 0$, the equation has two equal real roots.
3. If $D < 0$, the equation has no real roots (the roots are complex/imaginary).

Since $D = -31$, and $-31 < 0$, the discriminant is negative.

Step 4: Conclusion regarding the existence of real roots.
Because the discriminant is less than zero ($D < 0$), the square root of the discriminant would result in an imaginary number. Therefore, the quadratic equation $2x^2 - 3x + 5 = 0$ does not possess any real roots.

Final Answer: The discriminant is $-31$, which is less than $0$. Therefore, the given quadratic equation has no real roots.


More Questions from Class 10 Mathematics Quadratic Equations EXERCISE 4.3


CBSE Solutions for Class 10 Mathematics Quadratic Equations


Chapters in CBSE - Class 10 Mathematics


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Hi, I'm Ravi Shankar, a Mathematics educator with 10+ years of teaching experience. I help Class 10 students strengthen their concepts, improve application skills, and prepare systematically for important examinations. At this stage, simply knowing formulas isn't enough. Students need to recognise the appropriate method, interpret questions correctly, solve efficiently and avoid losing marks through preventable mistakes. My classes focus on: • Concept mastery • Algebra and functions • Geometry • Trigonometry • Statistics and probability • Application-based problems • Multi-step questions • Past-paper practice • Error analysis • Time management and exam strategy I use regular assessments to identify recurring mistakes and areas requiring additional attention. My approach: Concept → Application → Problem Solving → Exam Practice → Analysis The goal is to help students approach examinations with greater confidence, accuracy, and consistency. Book an introductory session to understand where your child currently stands and what needs to improve.

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I teach Mathematics to students of Classes 8 to 10, and I have been guiding young learners for over 25 years. My teaching method focuses on helping students develop not just strong academic results, but also a deeper confidence and love for the subject. I hold an M.Sc. in Mathematics and a B.Ed., which have given me both mastery of the subject and the pedagogical skills to teach it effectively. In addition to this, I am a certified Vedic Math trainer, which allows me to introduce students to faster calculation techniques and enhance their mental math abilities. Over the course of my career, I have taught multiple age groups and ability levels, always with a focus on clarity and practice. I firmly believe that every student is capable of excelling in math if the concepts are explained in a structured and relatable way. My students’ consistent success reflects my dedication and effective teaching style. I offer both online and offline classes. Each session is designed with a well-structured lesson plan, ensuring that every topic is covered in depth and revised frequently. I use interactive discussions to ensure active participation and encourage students to voice their doubts without hesitation. I make use of PPTs and a whiteboard during my lessons to visually explain concepts and illustrate multiple approaches to problem-solving. Along with concept explanation, I include plenty of guided practice and gradual difficulty progression to boost skills and confidence. Regular assessment, practice worksheets, revision tests, and personal attention are key elements of my approach. Yes, I also provide study materials and additional exercises to ensure clarity and practice. My goal is to make mathematics an enjoyable part of every student's learning journey.

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