NCERT Solutions class 10 maths chapter 4
Quadratic Equations
NCERT Solutions class 10 maths chapter 4 Exercise 4.1
Question 1.
Check whether the following are quadratic equations:
(i) (x + 1)
2 = 2(x – 3)
(ii) x
2 – 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3)
(iv) (x – 3)(2x +1) = x(x + 5)
(v) (2x – 1)(x – 3) = (x + 5)(x – 1)
(vi) x
2 + 3x + 1 = (x – 2)
2
(vii) (x + 2)
3 = 2x (x2 – 1)
(viii) x
3 – 4x
2 – x + 1 = (x – 2)
3
Answer
(i) Given
(x + 1)
2 = 2(x – 3)
(a+b)
2 = a
2+2ab+b
2
⇒ x
2 + 2x + 1 = 2x – 6
⇒ x
2 + 7 = 0
which is form of ax
2 + bx + c = 0. where b=0
It is a quadratic equation.
(ii) Given,
x
2 – 2x = (–2) (3 – x)
⇒ x
2 – 2x = -6 + 2x
⇒ x
2 – 4x + 6 = 0
which is form of ax
2 + bx + c = 0.
It is quadratic equation.
(iii) Given,
(x – 2)(x + 1) = (x – 1)(x + 3)
⇒ x
2 – x – 2 = x
2 + 2x – 3
Which is not in the form of ax2 + bx + c = 0.
It is not a quadratic equation.
(iv) Given,
(x – 3)(2x +1) = x(x + 5)
⇒ 2x2 – 5x – 3 = x2 + 5x
⇒ x2 – 10x – 3 = 0
which is form of ax2 + bx + c = 0.
It is quadratic equation.
(v) Given,
(2x – 1)(x – 3) = (x + 5)(x – 1)
⇒ 2x2 – 7x + 3 = x2 + 4x – 5
⇒ x2 – 11x + 8 = 0
It is form of ax2+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is quadratic equation.
(vi) Given,
x2 + 3x + 1 = (x – 2)2
⇒ x2 + 3x + 1 = x2 + 4 – 4x
⇒ 7x – 3 = 0
It is form of ax2 + bx + c = 0.
where a=0 and b,c are any real number
It is not a quadratic equation.
(vii) Given,
(x + 2)3 = 2x(x2 – 1)
⇒ x3 + 8 + x2 + 12x = 2x3 – 2x
⇒ x3 + 14x – 6x2 – 8 = 0
It is not in the form of ax2 + bx + c = 0.
It is not a quadratic equation.
(viii) Given, x3< – 4x2< – x + 1 = (x – 2)3<
x3< – 4x2< – x + 1 = x3< – 8 – 6x2< + 12x
2x2<– 13x + 9 = 0
It is the form of ax2+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is quadratic equation.
Question 2.
Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken
Answer 2
(i)let the
Breadth of the rectangular plot = x m
Thus, the length of the plot = (2x + 1) m
Area of rectangle = length × breadth
(2x + 1)x = 528
2x2 + x =528
2x2 + x – 528 = 0
which is of the form of ax2+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.
(ii) Let
The first integer number = x
Thus, the next consecutive positive integer will be = x + 1
According to the question
Product of two consecutive integers = x X (x +1) = 306
x2+ x = 306
x2 + x – 306 = 0
It is of the form of ax2+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.
.
(iii) Let
Age of Rohan’s = x years
Rohan’s mother’s age = x + 26
After 3 years,
Age of Rohan’s = x + 3
Age of Rohan’s mother will be = x + 26 + 3 = x + 29br>
According to question
(x + 3)(x + 29) = 360
x2 + 29x + 3x + 87 = 360
x2 + 32x + 87 – 360 = 0
x2 + 32x – 273 = 0
which is of the form of ax2+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.
(iv) Let us consider,
The speed of train = x km/h
Time taken to travel 480 km = 480/x km/hr
speed of train = (x – 8) km/h
Therefore, time taken to travel 480 km =
480
/
(x-8)
km/h
According to the question
Speed × Time = Distance
(
480
/
(x-8)
)
- (
480
/
x
)=3
(
480x-480(x-8)
/
x(x-8)
)=3
480x - 480x +3840 =3x(x-8)
3840= 3x2 - 24x
3x2 - 24x - 3840=0
x2– 8x – 1280 = 0
x2 – 8x – 1280 = 0
which is of the form of ax2+ bx + c = 0.
where a ≠ 0, and b,c are any real number
It is represent the required quadratic equation.
.