Class 10 Maths Chapter 4 Quadratic Equations – Exercise 4.3 NCERT Solutions

Introduction

Exercise 4.3 focuses on solving quadratic equations using the quadratic formula. This method is universal and works for all quadratic equations, even when factorization is not possible. It also introduces the discriminant, which helps determine the nature of roots.

Formula Used

• Quadratic Formula:

$$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

• Discriminant:

$$D=b^2-4ac$$

• If $D>0$: Two distinct real roots

• If $D=0$: Two equal real roots

• If $D<0$: No real roots (complex roots)

NCERT Questions with Solutions (10)

Q1. Solve $2x^2-7x+3=0$.

Solution: $a=2,b=-7,c=3$. $D=(-7{)}^2-4(2)(3)=49-24=25$. Roots: $\frac{7\pm 5}{4}$. $x=3,\frac{1}{2}$.

Q2. Solve $x^2-2x-3=0$.

Solution: $a=1,b=-2,c=-3$. $D=4+12=16$. Roots: $\frac{2\pm 4}{2}$. $x=3,-1$.

Q3. Solve $x^2+4x+4=0$.

Solution: $a=1,b=4,c=4$. $D=16-16=0$. Roots: $\frac{-4}{2}=-2$. Equal roots: $-2,-2$.

Q4. Solve $3x^2-5x+2=0$.

Solution: $a=3,b=-5,c=2$. $D=25-24=1$. Roots: $\frac{5\pm 1}{6}$. $x=1,\frac{2}{3}$.

Q5. Solve $x^2-6x+9=0$.

Solution: $a=1,b=-6,c=9$. $D=36-36=0$. Roots: $\frac{6}{2}=3$. Equal roots: 3, 3.

Q6. Solve $2x^2+x-6=0$.

Solution: $a=2,b=1,c=-6$. $D=1+48=49$. Roots: $\frac{-1\pm 7}{4}$. $x=\frac{6}{4}=\frac{3}{2},x=-2$.

Q7. Solve $x^2+2x+5=0$.

Solution: $a=1,b=2,c=5$. $D=4-20=-16$. No real roots (complex roots).

Q8. Solve $x^2-4x+3=0$.

Solution: $a=1,b=-4,c=3$. $D=16-12=4$. Roots: $\frac{4\pm 2}{2}$. $x=3,1$.

Q9. Solve $5x^2-6x+1=0$.

Solution: $a=5,b=-6,c=1$. $D=36-20=16$. Roots: $\frac{6\pm 4}{10}$. $x=1,\frac{1}{5}$.

Q10. Solve $x^2+x+1=0$.

Solution: $a=1,b=1,c=1$. $D=1-4=-3$. No real roots (complex roots).

FAQs (10 from NCERT)

1. Q: What is the quadratic formula? A: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

2. Q: What is the discriminant? A: $D=b^2-4ac$.

3. Q: What does discriminant tell us? A: Nature of roots (real, equal, or complex).

4. Q: What are real roots? A: Roots that are real numbers.

5. Q: What are complex roots? A: Roots that involve imaginary numbers.

6. Q: When do we get equal roots? A: When $D=0$.

7. Q: When do we get distinct real roots? A: When $D>0$.

8. Q: When do we get no real roots? A: When $D<0$.

9. Q: Why is quadratic formula useful? A: It works for all quadratic equations.

10. Q: Why is this exercise important? A: It builds algebraic skills and prepares for advanced mathematics.

Conclusion

Exercise 4.3 has 10 solved questions and 10 FAQs that strengthen your understanding of solving quadratic equations using the quadratic formula. This builds the foundation for advanced algebraic techniques in Class 10 Maths.

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