In this page we have NCERT book Solutions for Class 10th Maths: Chapter 4-Quadratic Equations for EXERCISE 4.1 on page number 73 and 74. Hope you like them and do not forget to like , social_share and comment at the end of the page.
Question 1
Check whether the following are quadratic equationsSolution 1
we know that
Quadratic equation
ax^{2} +bx+c =0 where a≠0
We know that
(a+b)^{2}=a^{2}+b^{2}+2ab
⇒x^{2} + 2x+1=2x-6
Simplifying it
⇒;x^{2} +7=0
Since it is of a quadratic form : ax^{2} +bx+c =0 where a≠0
with b=0
So it is a quadratic equation
x^{2} -2x=(-2)(3-x)
Simplifying it
x^{2} -2x=-6+2x
⇒x^{2} -4x+6=0
Since it is a quadratic form
ax^{2} +bx+c =0 where a≠0
So it is a quadratic equation
x^{2} -2x=(-2)(3-x)
(x-2)(x+1)=(x-1)(x+3)
Multiplying both the factors
⇒x^{2} -2x+2+x= x^{2} +3x-x-3
Simplifying
-3x+1=0
It is not of the quadratic form
ax^{2} +bx+c =0 where a≠0
So it is not a quadratic equation
(x – 3)(2x +1) = x(x + 5)
Multiplying both the factors
2x^{2}+x-6x-3=x^{2}+5x
Simplifying
x^{2} -10x-3=0
Since it is a quadratic form
ax^{2} +bx+c =0 where a≠0
⇒So it is a quadratic equation
(2x – 1)(x – 3) = (x + 5)(x – 1)
Multiplying both the factors on both sides
2x^{2} -6x-x+3=x^{2} –x+5x-5
⇒x^{2} -11x +8=0
Since it is a quadratic form
ax^{2} +bx+c =0 where a≠0
So it is a quadratic equation
x^{2} + 3x + 1 = (x – 2)^{2}
We know that
(a+b)^{2}=a^{2}+b^{2}+2ab
⇒x^{2} + 3x + 1 =x^{2}-4x+4
⇒7x-3=0
Since it is not of quadratic form
ax^{2} +bx+c =0 where a≠0
So it is a not quadratic equation
(x + 2)^{3} = 2x (x^{2} – 1)
Important formula you must have remembered in old classes
(a+b)^{3}= a^{3} +b^{3}+3ab^{2}+3a^{2}b
⇒x^{3} +8+6x^{2}+12x=2x^{3} -2x
Simplifying
x^{3}-6x^{2} -14x-8=0
Since it is not of quadratic form
ax^{2} +bx+c =0 where a≠0
So it is a not quadratic equation
x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}
Important formula you must have remembered in old classes
(a-b)^{3}= a^{3} -b^{3}+3ab^{2}-3a^{2}b
x^{3} – 4x^{2} – x + 1 =x^{3}-8-6x^{2}+12x
Simplifying
2x^{2} -13x+9=0
Since it is a quadratic form
ax^{2} +bx+c =0 where a≠0
So it is a quadratic equation
Question 2
Represent the following situations in the form of quadratic equations :
Solution
Let the breath of the plot= x m
As per given condition in the question
Length =2x+1
Now we know that Area is given by
A=LB
A=528 m^{2}
So
528=(2x+1)x
⇒ x^{2}+2x-528=0
Which is a quadratic equation
let the two consecutive positive integers are x and x+1
The product of these would be
x(x+1)
It is given that product is 306
So
⇒ x(x+1)=306
⇒ x^{2}+x-306=0
⇒ Which is a quadratic equation
Let Rohan present age=x year
Then Rohan Mother present age would =x+26
After 3 year,
Rohan age would be =x+3
Rohan mother’s age would be =x+26+3=x+29
According to question, The product of their ages (in years) =360
Then
(x+3)(x+29)= 360
Simplifying
⇒ x^{2} +29x +3x+87=360
⇒ x^{2} +32x -273=0
Which is a quadratic equation
Let the speed of the train is x km/hr
Now distance travelled by the train=480 km
Few important formula here
Speed=Distance/time
Or Time= Distance /Speed
Case I
Time taken to travel 480 km by train will be =480/x
Case II
Now the speed of the train is reduced by 8 km/hr,
So speed would (x-8)
Now Time taken to travel 480 km will be =480/x-8
Now as per the question
480/(x-8) - 480/x =3
⇒ [480x-480(x-8)]/x(x-8) =3
⇒ 480x-480x+3840=3x(x-8)
⇒ 3x^{2} -24x-3840=0
⇒ Which is a quadratic equation