Quadratic Equations - Class 10 CBSE Math Solutions of NCERT Mathematics - Chapter 4 Textbook Exercise 4.1

Solutions of CBSE Board, Class 10 Mathematics (SA-II / Term II)  

Chapter 4, Quadratic EquationS  

NCERT Solutions for Mathematics Textbook Exercise 4.1

(Page 73)

1:  Check whether the following are quadratic equations:

(i)  (x + 1)² = 2(x - 3)                           (ii)  x² - 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)  x² + 3x + 1 = (x - 2)²

(vii)  (x + 2)³ = 2x(x² - 1)                   (viii)  x³ -4x² - x + 1 = (x - 2)³

Solution.1:

(i) LHS = (x + 1)² = x² + 2x + 1

or,  x² + 2x + 1 = 2(x - 3)

or, x² + 2x + 1 = 2x - 6

or, x² + 7 = 0

The above expression is of the form: ax² + bx + c = 0

Therefore, the given equation is a quadratic equation.

(ii) LHS = x² - 2x

Given that,  x² - 2x = (–2) (3 - x)

or, x² - 2x - 2x + 6 = 0

or, x² - 4x + 6 = 0

It is of the form: ax² + bx + c = 0

Therefore, the given equation is a quadratic equation.

(iii) LHS = (x - 2) (x + 1) = x(x + 1) - 2(x + 1)

 or, LHS = x² - x -2,

Now as per given equation -

 x² - x -2 = (x - 1) (x + 3) = x(x + 3) - 1(x + 3)

or,  x² - x -2 = x² + 3x - x - 3

or,  –x - 2 - 2x + 3 = 0

or, –3x +1 = 0

It is not of the form: ax² + bx + c = 0

Therefore, the given equation is not a quadratic equation.

(iv), (v) & (vi) taking hints from above solutions try top solve your-self.

 (vii) LHS = (x + 2)³ = x³ + 2³ + 3 (x) (2) (x + 2)

or, LHS = x³ + 8 + 6x² + 12x

As per the given equation,

x³ + 6x² + 12x + 8 = 2x(x² - 1)

or, x³ + 6x² + 12x + 8 = 2x³ - 2x

or, –x³ + 6x² + 14x + 8 = 0

Since, it is not in the form of ax² + bx + c = 0

Therefore, the given equation is not a quadratic equation.

                  

(viii) Do it yourself.

     

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 meters) 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/hr less, then it would have taken 3 hrs more to cover the same distance. We need to find the speed of the train.      

Solution.2:

(i) Let the breadth be ‘x’.

As per the given condition,

Length = (2x + 1)

Now area of the rectangular plot = x (2x + 1) which is given 528 m²

or, x (2x + 1) = 528

or, 2x² + x - 528 = 0

Hence, the value of ‘x’ which satisfies the above quadratic equation 2x² + x - 528 = 0, is the breadth of the rectangular plot.

(ii) Let the first integer be ‘x’and the second integer be (x + 1).

As per the given condition,

x (x + 1) = 306

or, x² + x - 306 = 0

Hence, the value of ‘x’ which satisfies the above quadratic equation:  x² + x - 306 = 0, where ‘x’ is the smallest integer.

(iii) Let Rohan’s age be ‘x’.

His mother’s age = (x + 26)

After 3 years,

Rohan’s age = x + 3

Mother’s age = x + 26 + 3 = x + 29

As per the given condition, the product of their ages after 3 years is 360.

So, (x + 3) (x + 29) = 360

or, x² + 32x + 87 = 360

or, x² + 32x - 273 = 0

Hence, the value of ‘x’ which satisfies the above quadratic equation: x² + 32x - 273 = 0, is the present age of Rohan.

(iv) Taking hint from above solutions try to solve this yourself. 

At the end of completing all Exercises from each Chapter given in NCERT Math textbook, we shall provide CBSE Notes (Hints) / CBSE Solutions, Mathematics Sample Questions with their Solutions and many more…    

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