Rajasthan Board RBSE Solutions for Class 10 Maths Chapter 4 Quadratic Equations Ex 4.1 Textbook Exercise Questions and Answers.

## RBSE Class 10 Maths Solutions Chapter 4 Quadratic Equations 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 (x^{2} – 1)

(viii) x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}

Solution:

(i) According to the question

(x + 1)^{2} = 2(x – 3)

or x^{2} + 1 + 2x = 2x – 6

or x^{2} + 1 + 2x – 2x + 6 = 0

or x^{2} + 7 = 0

or x^{2} + 0x + 7 = 0

Which is an equation of the form ax^{2} + bx + c = 0. (a ≠ 0)

∴ It is a quadratic equation.

(ii) According to the question

x^{2} – 2x = (-2)(3 – x)

or x^{2} – 2x = -6 + 2x

or x^{2} – 2x + 6 – 2x = 0

or x^{2} – 4x + 6 = 0

Which is an equation of the form ax^{2} + bx + c = 0; (a ≠ 0)

∴ It is a quadratic equation.

(iii) According to the question

(x – 2) (x + 1) = (x – 1) (x + 3)

or x^{2} + x – 2x – 2 = x^{2} + 3x – x – 3

or x^{2} – x – 2 = x^{2} + 2x – 3

or x^{2} – x – 2 – x^{2} – 2x + 3 = 0

or – 3x + 1 = 0

Which contain no terms of x^{2}

∴ It is not a quadratic equation.

(iv) According to the question

(x – 3) (2x + 1) = x (x + 5)

or 2x^{2} + x – 6x – 3 = x^{2} + 5x

or 2x^{2} – 5x – 3 – x^{2} – 5x = 0

or x^{2} – 10x – 3 = 0

Which is an equation of the form ax^{2} + bx + c = 0; (a ≠ 0)

∴ It is a quadratic equation.

(v) According to the question

(2x – 1) (x – 3) = (x + 5) (x – 1)

or 2x^{2} – 6x – x + 3 = x^{2} – x + 5x – 5

or 2x^{2} – 7x + 3 = x^{2} + 4x – 5

or 2x^{2} – 7x + 3 – x^{2} – 4x + 5 = 0

or x^{2} – 11x + 8 = 0

Which is an equation of the form ax^{2} + bx + c = 0; (a ≠ 0)

∴ It is a quadratic equation.

(vi) According to the question

x^{2} + 3x + 1 = (x – 2)^{2}

or x^{2} + 3x + 1 = x^{2} + 4 – 4x

or x^{2} + 3x + 1 – x^{2} – 4 + 4x = 0

or 7x – 3 = 0

Which is an equation of the form ax^{2} + bx + c = 0; (a ≠ 0)

∴ It is a quadratic equation.

(vii) According to the question

(x + 2)^{3} = 2x(x^{2} – 1)

or x^{3} + (2)^{3} + 3(x)^{2} 2 + 3(x)(2)^{2} = 2x^{3} – 2x

or x^{3} + 8 + 6x^{2} + 12x = 2x^{3} – 2x

or x^{3} + 8 + 6x^{2} + 12x – 2x^{3} + 2x = 0

or -x^{3} + 6x^{2} + 14x + 8 = 0

Here the highest power of x is 3

Therefore it is a cubic equation.

∴ It is not a quadratic equation.

(viii) According to the question x^{3} – 4x^{2} – x + 1 = (x – 2)^{3}

or x^{3} – 4x^{2} – x + 1 = x^{3} – (2)^{3} + 3(x)^{2}(-2) + 3(x) (- 2)^{2}

or x^{3} – 4x^{2} – x + 1 = x^{3} – 8 – 6x^{2} + 12x

or x^{3} – 4x^{2} – x + 1 – x^{3} + 8 + 6x^{2} – 12x = 0

or 2x^{2} – 13x + 9 = 0

Which is an equation of the form ax^{2} + bx + c = 0; (a ≠ 0)

∴ It is a quadratic equation.

Question 2.

Represent the following situations in the form of quadratic equations :

(i) The area of a rectangular plot is 528 m^{2}. 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 3 hours more to cover the same distance. We need to find the speed of the train.

Solution:

(i) Let the breadth of the rectangular plot = x m

∴ Length of the rectangular plot = (2x + 1) m

∴ Area of the rectangular plot = [x(2x + 1)]m^{2}

= (2x^{2} + x) m^{2}

According to the question

2x^{2} + x = 528

or 2x^{2} + x – 528 = 0

Hence the quadratic equation is

2x^{2} + x – 528 = 0

Where x (in metres) is the breadth of the plot.

(ii) Let x and x + 1 be two consecutive positive integers.

Then,

Product of the integers = x (x + 1)

= x^{2} + x

According to the question

x^{2} + x = 306

or x^{2} + x – 306 = 0

Hence the given problem in the form of a quadratic equation is

x^{2} + x – 306 = 0

where x is the smaller integer.

(iii) Let Rohan’s present age= x years

Then, Age of Rohan’s mother = (x + 26) years

3 years from now,

Rohan’s age = (x + 3) years

and Age of Rohan’s mother = (x + 26 + 3) years

= (x + 29) years

Their product = (x + 3) (x + 29)

= x^{2} + 29x + 3x + 87

= x^{2} + 32x + 87

According to the question,

x^{2} + 32x + 87 = 360

or x^{2} + 32x + 87 – 360 = 0

or x^{2} + 32x – 273 = 0

Hence the given problem in the form of a quadratic equation is

x^{2} + 32x – 273 = 0

where x (in years) is the present age of Rohan.

(iv) Let the speed of the train be u km./h. Then,

time taken in covering a distance of 480 km = \(\frac{480}{u}\) hours [∵ time = \(\frac{Distance}{Speed}\)]

If the speed of the train had been 8 km/ h less, then time taken in covering the same distance = \(\frac{480}{u-8}\)hours

According to the question 3 hours are taken more to cover the same distance

Therefore \(\frac{480}{u-8}\) – \(\frac{480}{u}\) = 3

or \(\frac{160}{u-8}\) – \(\frac{160}{u}\) = 1

or 160u – 160 (u – 8) = u(u – 8)

or 160u – 160u + 1280 = u^{2} – 8u

or u^{2} – 8u – 1280 = 0

Hence the quadratic equation form of the given problem is

u^{2} – 8u – 1280 = 0

Where u (in km/h) is the speed of the train.

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