Class 10 Maths Chapter 4 Quadratic Equations – Exercise 4.4 NCERT Solutions

Introduction

Exercise 4.4 applies quadratic equations to real‑life word problems. You’ll learn how to form quadratic equations from practical situations such as age problems, geometry, and number puzzles, and then solve them using factorization or the quadratic formula.

Formula Used

• General Form of Quadratic Equation:

$$a{x}^2+bx+c=0$$

• Quadratic Formula:

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

• Factorization Method: Split middle term and factorize.

NCERT Questions with Solutions (10)

Q1. The product of two consecutive positive integers is 306. Find the integers.

Solution: Let integers = $x,x+1$. Equation: $x(x+1)=306\Rightarrow x^2+x-306=0$. Solve: Roots = 17, -18. Positive integers = 17, 18.

Q2. The sum of squares of two consecutive odd integers is 290. Find the integers.

Solution: Let integers = $x,x+2$. Equation: $x^2+(x+2{)}^2=290$. Simplify: $2x^2+4x+4=290\Rightarrow 2x^2+4x-286=0$. Solve: Roots = 11, -13. Odd integers = 11, 13.

Q3. A natural number, when increased by 12, equals 160 divided by the number. Find the number.

Solution: Equation: $x+12=\frac{160}{x}$. Multiply: $x^2+12x-160=0$. Solve: Roots = 8, -20. Natural number = 8.

Q4. The area of a rectangle is 528 m². Length is 2 m more than twice breadth. Find dimensions.

Solution: Let breadth = $x$. Length = $2x+2$. Equation: $x(2x+2)=528\Rightarrow 2x^2+2x-528=0$. Solve: $x=22$. Length = 46. Dimensions = 22 m × 46 m.

Q5. A train travels 360 km at a uniform speed. If speed had been 5 km/h more, it would have taken 48 minutes less. Find speed.

Solution: Let speed = $x$. Time = $\frac{360}{x}$. Equation: $\frac{360}{x}-\frac{360}{x+5}=\frac{48}{60}=\frac{4}{5}$. Simplify: $360(5)=\frac{4}{5}x(x+5)$. Solve: $x=45$. Speed = 45 km/h.

Q6. The difference of squares of two numbers is 180. If sum is 30, find numbers.

Solution: Let numbers = $x,y$. Equation: $x+y=30$, $x^2-y^2=180$. Factorize: $(x-y)(x+y)=180\Rightarrow (x-y)(30)=180\Rightarrow x-y=6$. Solve: $x=18,y=12$.

Q7. A motorboat goes 16 km downstream in 2 hours, and returns upstream in 4 hours. Find speed of boat in still water and speed of stream.

Solution: Let boat speed = $x$, stream speed = $y$. Downstream: $\frac{16}{x+y}=2\Rightarrow x+y=8$. Upstream: $\frac{16}{x-y}=4\Rightarrow x-y=4$. Solve: $x=6,y=2$.

Q8. The sum of ages of father and son is 55 years. Father’s age is 5 years more than twice son’s age. Find ages.

Solution: Let son’s age = $x$. Father’s age = $2x+5$. Equation: $x+2x+5=55\Rightarrow 3x+5=55$. Solve: $x=50\mathrm{/}3\approx 16.7$. Father ≈ 33.3.

Q9. A two‑digit number is such that product of digits is 24. If 18 is added, digits interchange. Find number.

Solution: Let digits = $x,y$. Number = $10x+y$. Equation: $xy=24$. Also: $10x+y+18=10y+x$. Simplify: $9x-9y=-18\Rightarrow x-y=-2$. Solve: $x=4,y=6$. Number = 46.

Q10. A rectangular park is 60 m long and 40 m wide. A path of equal width runs inside along the boundary. Area of path is 476 m². Find width.

Solution: Let width = $x$. Inner rectangle = $(60-2x)(40-2x)$. Area of path = $2400-(60-2x)(40-2x)=476$. Simplify: $2400-(2400-200x+4x^2)=476\Rightarrow 200x-4x^2=476$. Equation: $4x^2-200x+476=0$. Solve: $x=2$. Width = 2 m.

FAQs (10 from NCERT)

1. Q: What is a quadratic word problem? A: A real‑life situation modeled by a quadratic equation.

2. Q: How do you form quadratic equations from problems? A: Translate conditions into algebraic expressions.

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

4. Q: What is factorization method? A: Splitting middle term to factorize quadratic.

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

6. Q: What does discriminant tell us? A: Nature of roots.

7. Q: Why are quadratic word problems important? A: They connect maths to real‑life situations.

8. Q: Can quadratic equations have equal roots? A: Yes, if $D=0$.

9. Q: Can quadratic equations have no real roots? A: Yes, if $D<0$.

10. Q: Why is this exercise important? A: It builds problem‑solving skills and practical applications.

Conclusion

Exercise 4.4 has 10 solved questions and 10 FAQs that strengthen your understanding of solving real‑life word problems using quadratic equations. This builds the foundation for applying algebra to practical scenarios in Class 10 Maths.

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