Mastering Quadratic Equations Class 10 – Smart Learning with FUZY MATH ACADEMY

student taking online class

✍️Mastering Quadratic Equations Class 10 – Smart Learning with FUZY MATH ACADEMY

Introduction

Mathematics has always been a subject of curiosity, logic, and problem-solving. Among all algebraic concepts in Class 10, the chapter on Quadratic Equations (Chapter 4, NCERT) is considered one of the most important. Not only does it carry high weightage in board exams, but it also plays a key role in competitive exams like NTSE, Olympiads, and later in JEE/NEET preparation.

At FUZY MATH ACADEMY, we believe that students from Classes 5 to 12 deserve the best online learning experience powered by an AI-enabled LMS and 24/7 chatbot support. With our smart teaching methods, live + recorded lectures, and instant doubt-solving tools, learning Quadratic Equations becomes simple, effective, and engaging.

In this blog, let’s understand Quadratic Equations step-by-step, with definitions, formulas, applications, and 20 important NCERT-based questions with solutions.

📑What is a Quadratic Equation?

A quadratic equation in one variable is an equation of the form:

$$a{x}^2+bx+c=0(a\mathrm{\ne }0)$$

Where:

• $a,b,c$ are real numbers

• $a\mathrm{\ne }0$ (since otherwise it becomes a linear equation)

Example:

$$2x^2 - 5x + 3 = 0,$$

👉Methods of Solving Quadratic Equations

1. Factorization Method
   Break the middle term and solve by splitting into two linear factors.

2. Completing the Square Method
   Convert the equation into a perfect square and then solve.

3. Quadratic Formula Method
   Using the formula:

   $$x=\frac{-b\pm \sqrt{b^2-4a\mathrm{c/}}}{2a}$$

Here, the term $b^2-4ac$ is called the Discriminant (D).

• If $D>0$ 2 distinct real roots

• If $D=0$ 2 equal real roots

• If $D<0$: No real roots

🔍Real-Life Applications of Quadratic Equations

• Calculating areas of plots

• Projectile motion problems in physics

• Maximum profit and minimum cost in business

• Motion problems (time & distance)

• Designing architecture and bridges

📌Why Students Struggle with Quadratic Equations

• Confusion in formula application

• Mistakes in sign while solving

• Lack of practice in word problems

• Not understanding discriminant properly

💬At FUZY MATH ACADEMY, these challenges are solved with:

• Step-by-step LMS tutorials

• Instant practice quizzes

• AI-chatbot for immediate doubt solving

• Real-world examples to boost clarity

💎20 Important NCERT-Based Questions with Solutions

Quadratic Equations – Important Questions with Solutions

Q1. Solve: x2 − 5x + 6 = 0

Step 1: Try factorization: x2 − 5x + 6 = (x−2)(x−3).

Step 2: Set each factor zero: x−2=0 ⇒ x=2 and x−3=0 ⇒ x=3.

Answer: x = 2, 3

Q2. Solve: 2x2 − 7x + 3 = 0

Step 1: Compute discriminant: D = b2 − 4ac = (−7)2 − 4·2·3 = 49 − 24 = 25.

Step 2: Use quadratic formula: x = [7 ± √25] / (2·2) = (7 ± 5)/4.

Answer: x = 3 or x = 1/2.

Q3. Solve: x2 − 2x − 15 = 0

Step 1: Factorize: find two numbers whose product = −15 and sum = −2 → −5 and 3.

Step 2: x2 − 5x + 3x − 15 = (x−5)(x+3).

Answer: x = 5, −3.

Q4. Solve using quadratic formula: x2 + 4x + 5 = 0

Step 1: D = 42 − 4·1·5 = 16 − 20 = −4.

Step 2: Since D < 0, no real roots (roots are complex).

Answer: No real roots (complex conjugates).

Q5. Product of two consecutive integers is 306. Find the integers.

Step 1: Let smaller integer = n. Then n(n+1)=306.

Step 2: n2 + n − 306 = 0. Solve using quadratic formula or factorization.

Discriminant: 1 + 4·306 = 1 + 1224 = 1225 = 352.

So n = [−1 ± 35]/2 ⇒ positive root n = (34)/2 = 17.

Answer: Integers are 17 and 18.

Q6. Solve: 3x2 − 5x + 2 = 0

Step 1: Discriminant D = (−5)2 − 4·3·2 = 25 − 24 = 1.

Step 2: x = [5 ± 1] / (2·3) = (5 ± 1)/6 ⇒ x = 1 or x = 2/3.

Answer: x = 1, 2/3.

Q7. Solve: 2x2 − 4x + 2 = 0

Step 1: D = (−4)2 − 4·2·2 = 16 − 16 = 0.

Step 2: Repeated root: x = 4 / (2·2) = 4/4 = 1.

Answer: x = 1 (equal roots).

Q8. Sum of squares of two consecutive integers is 365. Find the integers.

Step 1: Let smaller = n then n2 + (n+1)2 = 365.

Step 2: Simplify: n2 + n2 + 2n + 1 = 365 ⇒ 2n2 + 2n − 364 = 0 ⇒ n2 + n − 182 = 0.

Step 3: Discriminant 1 + 728 = 729 = 272. So n = (−1 ± 27)/2 ⇒ positive root n = 13.

Answer: Integers are 13 and 14.

Q9. Solve: x2 − 8x + 12 = 0

Step 1: Factorize: (x−6)(x−2) = 0.

Answer: x = 6, 2.

Q10. Solve: 6x2 − 11x − 35 = 0

Step 1: D = (−11)2 − 4·6·(−35) = 121 + 840 = 961 = 312.

Step 2: x = [11 ± 31] / (2·6) = (11 ± 31) / 12. So x = 42/12 = 7/2 or x = −20/12 = −5/3.

Answer: x = 7/2, −5/3.

Q11. The area of a rectangle is 54 sq. units. Its length is 6 more than its breadth. Find dimensions.

Step 1: Let breadth = b, length = l = b + 6. Then b(b+6)=54.

Step 2: b2 + 6b − 54 = 0. Discriminant D = 62 − 4·1·(−54) = 36 + 216 = 252.

Step 3: √252 = √(36·7) = 6√7. So b = [−6 ± 6√7]/2 = −3 ± 3√7. Take positive: b = −3 + 3√7.

Then l = b + 6 = 3 + 3√7 (approx values: b ≈ 4.937, l ≈ 10.937).

Answer: b = −3 + 3√7, l = 3 + 3√7.

Q12. Solve: x2 + 7x + 12 = 0

Step 1: Factorize: (x+3)(x+4) = 0.

Answer: x = −3, −4.

Q13. Solve: x2 − 2x − 8 = 0

Step 1: Factorize: (x−4)(x+2)=0.

Answer: x = 4, −2.

Q14. One root of a quadratic is double the other and sum of roots is 9. Find the roots and the quadratic equation.

Step 1: Let roots be r and 2r. Sum = 3r = 9 ⇒ r = 3.

Step 2: Roots are 3 and 6. For quadratic with these roots: sum = 9 ⇒ coefficient relation −b/a = 9 and product = 18 ⇒ c/a = 18.

Take a = 1 → equation: x2 − 9x + 18 = 0.

Answer: Roots: 3, 6. Equation: x2 − 9x + 18 = 0.

Q15. Solve: 2x2 − 9x + 7 = 0

Step 1: D = (−9)2 − 4·2·7 = 81 − 56 = 25.

Step 2: x = [9 ± 5]/(4) ⇒ x = 14/4 = 7/2 or x = 4/4 = 1.

Answer: x = 1, 7/2.

Q16. A train travels 360 km at speed x km/h. If speed were 5 km/h more, journey would take 48 minutes less. Find x.

Step 1: Time difference: 360/x − 360/(x+5) = 48 min = 48/60 = 4/5 hours.

Step 2: Multiply both sides by x(x+5): 360(x+5) − 360x = (4/5) x(x+5) ⇒ left = 360·5 = 1800.

So 1800 = (4/5) x(x+5) ⇒ multiply by 5: 9000 = 4x(x+5) ⇒ x2 + 5x − 2250 = 0.

Step 3: Discriminant D = 25 + 9000 = 9025 = 952 ⇒ x = [−5 + 95]/2 = 90/2 = 45 (positive root).

Answer: x = 45 km/h.

Q17. Solve: x2 − 10x + 25 = 0

Step 1: Recognize perfect square: (x−5)2 = 0.

Answer: x = 5 (repeated root).

Q18. Product of two consecutive odd numbers is 255. Find them.

Step 1: Let the smaller odd be n (odd). Then n(n+2)=255.

Step 2: n2 + 2n − 255 = 0. Discriminant D = 4 + 1020 = 1024 = 322.

Step 3: n = [−2 ± 32]/2 ⇒ positive root n = 15.

Answer: Numbers are 15, 17.

Q19. Solve: 4x2 − 4√3 x + 3 = 0

Step 1: Compute discriminant: D = (−4√3)2 − 4·4·3 = 48 − 48 = 0.

Step 2: Equal roots: x = (4√3)/(2·4) = √3/2.

Answer: x = √3 / 2 (double root).

Q20. If roots of ax2 + bx + c = 0 are equal, prove b2 = 4ac.

Step 1: For equal roots, discriminant D = b2 − 4ac = 0 by definition.

Step 2: Therefore b2 = 4ac. (This is exactly the condition for equal roots.)

Answer: b2 = 4ac (proved).

⌛Why Choose FUZY MATH ACADEMY for Quadratic Equations?

• AI-powered LMS with instant quizzes

• 24/7 Chatbot for doubt solving

• Experienced faculty with shortcuts & tricks

• Affordable fees compared to offline coaching

• Progress tracking and personalized dashboard

💥Conclusion

Quadratic Equations are not just an academic topic but a life skill that develops reasoning and problem-solving ability. With proper guidance and regular practice, this chapter can fetch full marks in Class 10 board exams.

At FUZY MATH ACADEMY, we simplify learning with technology-driven teaching, round-the-clock support, and affordable plans. Whether you’re in Class 5 building foundations or in Class 12 aiming for competitive exams, our platform ensures you always stay ahead.

📞 Contact: 6264302661

Quadratic Equations – Important Questions with Solutions

Q1. Solve: x² − 5x + 6 = 0

Solution: (x − 2)(x − 3) = 0 ⇒ x = 2, 3

Q2. Solve: 2x² − 7x + 3 = 0

Solution: D = 49 − 24 = 25 ⇒ x = (7 ± 5)/4 ⇒ x = 3, 1/2

Q3. Solve: x² − 2x − 15 = 0

Solution: (x − 5)(x + 3) = 0 ⇒ x = 5, −3

Q4. Solve using quadratic formula: x² + 4x + 5 = 0

Solution: D = 16 − 20 = −4 ⇒ No real roots

Q5. If product of two consecutive integers = 306, find them.

n² + n − 306 = 0 ⇒ n = 17, −18 ⇒ Integers = 17, 18

🌐 Visit: www.fuzymathacademy.com

👉YOU MAY ALSO LIKE THIS BLOG👇

FUZY Math Academy – Online Math Tuition for Classes 5 to 12

🚀 Enroll today with FUZY MATH ACADEMY and make Maths your strongest subject!

YouTube Facebook Instagram Twitter (X) Pinterest