Check whether the following are quadratic equations:

(i) (*x* + 1)^{2} = 2(*x* - 3)

(ii) *x*^{2} - 2*x* = (-2)(3 - *x*)

(iii) (*x* - 2)(*x* + 1) = (*x* - 1)(*x* + 3)

(iv) (*x* - 3)(2*x* + 1) = *x*(*x* + 5)

(v) (2*x* - 1)(*x* - 3) = (*x* + 5)(*x* - 1)

(vi) *x*^{2} + 3*x* + 1 = (*x* - 2)^{2}

(vii) (*x* + 2)^{3} = 2*x*(*x*^{2} - 1)

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

**Soln. :** (i) *x*^{2} + 2*x* + 1 = 2*x* - 6

⇒ *x*^{2} + 2*x* + 1 - 2*x* + 6 = 0

⇒ *x*^{2} + 7 = 0

Since *x*^{2} + 7 is a quadratic polynomial.

∴ (*x* + 1)^{2} = 2(*x* - 3) is a quadratic equation.

(ii) *x*^{2} - 2*x* = - 6 + 2*x* ⇒ *x*^{2} - 2*x* - 2*x* + 6 = 0

⇒ *x*^{2} - 4*x* + 6 = 0

Since *x*^{2} - 4*x* + 6 is a quadratic polynomial

∴ *x*^{2} - 2*x* = (-2)(3 - *x*) is a quadratic equation.

(iii) *x*^{2} - *x* - 2 = *x*^{2} + 2*x* - 3

⇒ *x*^{2} - *x* - 2 - *x*^{2} - 2*x* + 3 = 0

⇒ - 3*x* + 1 = 0

Since - 3*x* + 1 is a linear polynomial.

∴ (*x* - 2)(*x* + 1) = (*x* - 1)(*x* + 3) is not a quadratic equation.

(iv) 2*x*^{2} + *x* - 6*x* - 3 = *x*^{2} + 5*x*

⇒ 2*x*^{2} - 5*x* - 3 - *x*^{2} - 5*x* = 0 ⇒ *x*^{2} - 10*x* - 3 = 0

Since *x*^{2} - 10*x* - 3 is a quadratic polynomial.

∴ (*x* - 3)(2*x* + 1) = *x*(*x* + 5) is a quadratic equation.

(v) 2*x*^{2} - 6*x* - *x* + 3 = *x*^{2} - *x* + 5*x* - 5

⇒ 2*x*^{2} - *x*^{2} - 6*x* - *x* + *x* - 5*x* + 3 + 5 = 0

⇒ *x*^{2} - 11*x* + 8 = 0

Since *x*^{2} - 11*x* + 8 is a quadratic polynomial

∴ (2*x* - 1)(*x* - 3) = (*x* + 5)(*x* - 1) is a quadratic equation.

(vi) *x*^{2} + 3*x* + 1 = *x*^{2} - 4*x* + 4

⇒ *x*^{2} + 3*x* + 1 - *x*^{2} + 4*x* - 4 = 0 ⇒ 7*x* - 3 = 0

Since 7*x* - 3 is a linear polynomial.

∴ *x*^{2} + 3*x* + 1 = (*x* - 2)^{2} is not a quadratic equation.

(vii) *x*^{3} + 3*x*^{2}(2) + 3*x*(2)^{2} + (2)^{3} = 2*x*^{3} - 2*x*

⇒ *x*^{3} + 6*x*^{2} + 12*x* + 8 = 2*x*^{3} - 2*x*

⇒ *x*^{3} + 6*x*^{2} + 12*x* + 8 - 2*x*^{3} + 2*x* = 0

⇒ -*x*^{3} + 6*x*^{2} + 14*x* + 8 = 0

Since -*x*^{3} + 6*x*^{2} + 14*x* + 8 is a polynomial of degree 3

∴ (*x* + 2)^{3} = 2*x*(*x*^{2} - 1) is not a quadratic equation.

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

⇒ *x*^{3} - 4*x*^{2} - *x* + 1 = *x*^{3} - 6*x*^{2} + 12*x* - 8

⇒ *x*^{3} - 4*x*^{2} - *x* + 1 - *x*^{3} + 6*x*^{2} - 12*x* + 8 = 0

⇒ 2*x*^{2} - 13*x* + 9 = 0

Since 2*x*^{2} - 13*x* + 9 is a quadratic polynomial.

∴ *x*^{3} - 4*x*^{2} - *x* + 1 = (*x* - 2)^{3} is a quadratic equation.

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

**Soln. :** (i) Let the breadth = *x* metres

∴ Length = 2(Breadth) + 1

∴ Length = (2*x* + 1)metres

Since Length × Breadth = Area

∴ (2*x* + 1) × *x* = 528 ⇒ 2*x*^{2} + *x* = 528

⇒ 2*x*^{2} + *x* - 528 = 0

Thus, the required quadratic equation is

2*x*^{2} + *x* - 528 = 0

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

∴ Product of the positive integers = 306

∴ *x*(*x* + 1) = 306 ⇒ *x*^{2} + *x* = 306

⇒ *x*^{2} + *x* - 306 = 0

Thus, the required quadratic equation is *x*^{2} + *x* - 306 = 0

(iii) Let the present age of Rohan be *x* years

∴ Mother's age = (*x* + 26) years

After 3 years, Rohan's age = (*x* + 3) years.

Mother's age = [(*x* + 26) + 3] years = (*x* + 29) years

According to the condition, [Product of their ages after 3 years] = 360

⇒ (*x* + 3) × (*x* + 29) = 360

⇒ *x*^{2} + 29*x* + 3*x* + 87 = 360

⇒ *x*^{2} + 29*x* + 3*x* + 87 - 360 = 0

⇒ *x*^{2} + 32*x* - 273 = 0

Thus, the required quadratic equation is *x*^{2} + 32*x* - 273 = 0.

(iv) Let the speed of the train = *u* km/hr

Distance covered = 480 km

Time taken = Distance ÷ Speed

= (480 ÷ *u*) hours

In second case, Speed = (*u* - 8) km/hour

∴

According to the condition,

⇒ 480*u* - 480(*u* - 8) = 3*u*(*u* - 8)

⇒ 480*u* - 480*u* + 3840 = 3*u*^{2} - 24*u*

⇒ 3840 - 3*u*^{2} + 24*u* = 0 ⇒ *u*^{2} - 8*u* - 1280 = 0

Thus, the required quadratic equation is *u*^{2} - 8*u* - 1280 = 0.

Find the roots of the following quadratic equations by factorisation

(i) *x*^{2} - 3*x* - 10 = 0 (ii) 2*x*^{2} + *x* - 6 = 0

(iii)

(iv)

(v) 100*x*^{2} - 20*x* + 1 = 0

**Soln. :** (i) We have, *x*^{2} - 3*x* - 10 = 0,
*x*^{2} - 5*x* + 2*x* - 10 = 0

⇒ *x*(*x* - 5) + 2(*x* - 5) = 0 ⇒ (*x* - 5)(*x* + 2) = 0

Either *x* - 5 = 0 ⇒ *x* = 5 or *x* + 2 = 0 ⇒ *x* = -2

Thus, the required roots are *x* = 5 and *x* = -2

(ii) We have, 2*x*^{2} + *x* - 6 = 0

⇒ 2*x*^{2} + 4*x* - 3*x* - 6 = 0

⇒ 2*x*(*x* + 2) - 3(*x* + 2) = 0

⇒ (*x* + 2)(2*x* - 3) = 0

Either *x* + 2 = 0

⇒ *x* = -2 or 2*x* - 3 = 0 ⇒

Thus, the required roots are *x* = -2 and

(iii) We have

⇒

⇒

⇒

⇒

Either

⇒

Thus, the required roots are

(iv) We have,

⇒ 16*x*^{2} - 8*x* + 1 = 0 ⇒ 16*x*^{2} - 4*x* - 4*x* + 1 = 0

⇒ 4*x*(4*x* - 1) - 1(4*x* - 1) = 0

⇒ (4*x* - 1)(4*x* - 1) = 0

⇒

Thus, the required roots are

(v) We have, 100*x*^{2} - 20*x* + 1 = 0

⇒ 100*x*^{2} - 10*x* - 10*x* + 1 = 0

⇒ 10*x*(10*x* - 1) - 1(10*x* - 1) = 0

⇒ (10*x* - 1)(10*x* - 1) = 0

⇒ (10*x* - 1) = 0 and (10*x* - 1) = 0

⇒

Thus, the required roots are

Solve the problems :

(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was 750. We would like to find out the number of toys produced on that day.

**Soln. :** (i) Let John had *x* marbles and the Jivanti had (45 - *x*).

When both of them lost 5 marbles then equation becomes

∴ (*x* - 5) × (45 - *x* - 5) = 124

We have : *x*^{2} - 45*x* + 324 = 0

⇒ *x*^{2} - 9*x* - 36*x* + 324 = 0

⇒ *x*(*x* - 9) - 36(*x* - 9) = 0

⇒ (*x* *-* 9)(*x* - 36) = 0

Either *x* - 9 = 0 ⇒ *x* = 9 or *x* - 36 = 0 ⇒ *x* = 36

Thus, *x* = 9 and *x* = 36

∴ John has 9 marbles and Jivanti had

45 - 9 = 36 marbles

(ii) Let the number of toys produced on that day be *x*.

Then cost of 1 toy

⇒ ⇒ *x*^{2} - 55*x* = -750

⇒ *x*^{2} - 55*x* + 750 = 0

We have : *x*^{2} - 55*x* + 750 = 0

[∴ (-30) × (-25) = 750 and (-30) + (-25) = -55]

⇒ *x*^{2} - 30*x* - 25*x* + 750 = 0

⇒ *x*(*x* - 30) - 25(*x* - 30) = 0

⇒ (*x* - 30)(*x* - 25) = 0

Either *x* - 30 = 0 ⇒ *x* = 30

or *x* - 25 = 0 ⇒ *x* = 25

Thus, *x* = 30 and *x* = 25.

Find two numbers whose sum is 27 and product is 182.

SOLUTION:
**Soln. :** Here, sum of the numbers is 27.

Let one of the numbers be *x*.

∴ Other number = 27 - *x*

According to the condition, Product of the numbers = 182

⇒ *x*(27 - *x*) = 182 ⇒ 27*x* - *x*^{2} = 182

⇒ *x*^{2} - 27*x* + 182 = 0

⇒ *x*^{2} - 13*x* - 14*x* + 182 = 0

⇒ *x*(*x* - 13) - 14(*x* - 13) = 0

⇒ (*x* - 13)(*x* - 14) = 0

Either *x* - 13 = 0 ⇒ *x* = 13

or *x* - 14 = 0 ⇒ x = 14

Thus, the required numbers are 13 and 14.

Find two consecutive positive integers, sum of whose squares is 365.

SOLUTION:
**Soln. :** Let the two consecutive positive integers be *x* and (*x* + 1)

Since the sum of the square of the numbers

= 365

∴ *x*^{2} + (*x* + 1)^{2} = 365

⇒ *x*^{2} + [*x*^{2} + 2*x* + 1] = 365

⇒ *x*^{2} + *x*^{2} + 2*x* + 1 = 365

⇒ 2*x*^{2} + 2*x* + 1 - 365 = 0

⇒ 2*x*^{2} + 2*x* - 364 = 0 ⇒ *x*^{2} + *x* - 182 = 0

⇒ *x*^{2} + 14*x* - 13*x* - 182 = 0

⇒ *x*(*x* + 14) - 13(*x* + 14) = 0

⇒ (*x* + 14)(*x* - 13) = 0

Either *x* + 14 = 0 ⇒ *x* = -14 or *x* - 13 = 0

⇒ *x* = 13

Since *x* has to be a positive integer

⇒ *x* = -14 is rejected.

∴ *x* = 13 ⇒ *x* + 1 = 13 + 1 = 14

Thus, the required consecutive positive integers are 13 and 14.

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

SOLUTION:**Soln. :** Let the base of the given right triangle be '*x*' cm

∴ Its height = (*x* - 7) cm

∴

[By Pythagoras theorem]

∴

Squaring both sides, we get 169 = *x*^{2} + (*x* - 7)^{2}

⇒ 169 = *x*^{2} + *x*^{2} - 14*x* + 49

⇒ 2*x*^{2} - 14*x* + 49 - 169 = 0

⇒ 2*x*^{2} - 14*x* - 120 = 0

⇒ *x*^{2} - 7*x* - 60 = 0 ⇒ *x*^{2} - 12*x* + 5*x* - 60 = 0

⇒ *x*(*x* - 12) + 5(*x* - 12) = 0

⇒ (*x* - 12)(*x* + 5) = 0

Either *x* - 12 = 0 ⇒ *x* = 12

or *x* + 5 = 0 ⇒ *x* = -5

But the sides of a triangle can never be negative ⇒ *x* = -5 is rejected.

⇒ *x* = 12

∴ Length of the base = 12 cm

⇒ Length of the height = (12 - 7)cm = 5 cm

Thus, the required base = 12 cm and

height = 5 cm.

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was 90, find the number of articles produced and the cost of each article.

SOLUTION:**Soln. :** Let the number of articles produced in a day = *x*

∴ Cost of production of each article

= (2*x* + 3)

According to the condition, Total cost = 90

⇒ *x* × (2*x* + 3) = 90 ⇒ 2*x*^{2} + 3*x* = 90

⇒ 2*x*^{2} + 3*x* - 90 = 0 ⇒ 2*x*^{2} - 12*x* + 15*x* - 90 = 0

⇒ 2*x*(*x* - 6) + 15(*x* - 6) = 0

⇒ (*x* - 6)(2*x* + 15) = 0

Either *x* - 6 = 0 ⇒ *x* = 6

or 2*x* + 15 = 0 ⇒

But the number of articles produced can never be negative,

∴ is rejected ⇒ *x* = 6

∴ Cost of each article = (2 × 6 + 3) = 15

Thus, the required number of articles produced is 6 and the cost of each article is 15.

Find the roots of the following quadratic equations, if they exist, by the method of completing the square :

(i) 2*x*^{2} - 7*x* + 3 = 0 (ii) 2*x*^{2} + *x* - 4 = 0

(iii)

(iv) 2*x*^{2} + *x* + 4 = 0

**Soln. :** (i) We have, 2*x*^{2} - 7*x* + 3 = 0

Dividing throughout by the co-efficient of *x*^{2}, we get

⇒

⇒

⇒

⇒

⇒

⇒

Case I : When is positive,

then

⇒

Case II : When is negative,

then ⇒

Thus, required roots are *x* = 3, and

(ii) We have, 2*x*^{2} + *x* - 4 = 0

Dividing throughout by 2,

⇒

⇒

⇒

⇒

⇒

Case I : When is positive,

then

⇒

Case II : When is negative,

then

⇒

Thus, the required roots are

(iii) We have

Dividing throughout by 4, we have

Dividing throughout by 2, we have

But the square of a number cannot be negative.

∴ cannot give a real value.

⇒ There is no real value of *x* satisfy the given equation.

Find the roots of the following quadratic equations, using the quadratic formula :

(i) 2*x*^{2} - 7*x* + 3 = 0 (ii) 2*x*^{2} + *x* - 4 = 0

(iii) (iv) 2*x*^{2} + *x* + 4 = 0

**Soln. :** (i) Comparing the given equation with *ax*^{2} + *bx* + *c* = 0, we have
*a* = 2, *b* = -7, *c* = 3

∴ *b*^{2} - 4*ac* = (-7)^{2} - 4(2)(3) = 49 - 24 = 25 ≥ 0

Since *b*^{2} - 4*ac* > 0

∴ The given equation has real roots. The roots

(ii) Comparing the given equation with
*ax*^{2} + *bx* + *c* = 0 we have *a* = 2, *b* = 1, *c* = - 4

∴ *b*^{2} - 4*ac* = (1)^{2} - 4(2)(-4) = 1 + 32 = 33 > 0

Since *b*^{2} - 4*ac* > 0

∴ The given equation has real roots. The roots

(iii) Comparing the given equation with
*ax*^{2} + *bx* + *c* = 0, we have *a* = 4, *b* = , *c* = 3

∴ *b*^{2} - 4*ac* = ()^{2} - 4(4)(3)

= [16 × 3] - 48 = 48 - 48 = 0

Since *b*^{2} - 4*ac* = 0

∴ The given equation has real and equal roots, which are given by

⇒

∴

(iv) Comparing the given equation with
*ax*^{2} + *bx* + *c* = 0, we have *a* = 2, *b* = 1, *c* = 4

∴ *b*^{2} - 4*ac* = (1)^{2} - 4(2)(4) = 1 - 32 = -31 < 0

Since *b*^{2} - 4*ac* is less than 0, therefore the given equation does not have real roots.

Find the roots of the following equations :

(i)

(ii)

**Soln. :** (i) Here, we have :

⇒ *x*^{2} - 1 = -3*x* ⇒ *x*^{2} - 3*x* - 1 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0,

we have *a* = 1, *b* = -3, *c* = -1

∴ *b*^{2} - 4*ac* = (-3)^{2} - 4(1)(-1) = 9 + 4 = 13 > 0

∴

⇒

Now, taking positive sign,

Now, taking negative sign,

Thus, the required roots of the given equation are

(ii) We have

⇒

⇒

⇒ -11 × 30 = 11(*x*^{2} - 3*x* - 28)

⇒ - 30 = *x*^{2} - 3*x* - 28

⇒ *x*^{2} - 3*x* - 28 + 30 = 0

⇒ *x*^{2} - 3*x* + 2 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0,

we have *a* = 1, *b* = -3, *c* = 2

∴ *b*^{2} - 4*ac* = (-3)^{2} - 4(1)(2) = 9 - 8 = 1 > 0

∴ The given equation (1) has real roots, which are given by

⇒

Taking positive sign,

Taking negative sign,

Thus, the required roots are *x* = 2 and *x* = 1.

The sum of the reciprocals of Rehman's ages, (in years) 3 years ago and 5 years from now is Find his present age.

SOLUTION:**Soln. :** Let the present age of Rehman = *x*

∴ 3 years ago Rehman's age = (*x* - 3) years

5 years later Rehman's age = (*x* + 5) years

Now, according to the condition,

⇒

⇒ 3[*x* + 5 + *x* - 3] = (*x* - 3)(*x* + 5)

⇒ 3[2*x* + 2] = *x*^{2} + 2*x* - 15

⇒ 6*x* + 6 = *x*^{2} + 2*x* - 15

⇒ *x*^{2} + 2*x* - 6*x* - 15 - 6 = 0

⇒ *x*^{2} - 4*x* - 21 = 0 …(1)

Now, comparing (1) with *ax*^{2} + *bx* + *c* = 0, we have *a* = 1, *b* = -4, *c* = -21

∴ *b*^{2} - 4*ac* = (-4)^{2} - 4(1)(-21) = 16 + 84 = 100

Since,

⇒

Taking positive sign,

we have

Taking negative sign,

we have

Since age cannot be negative,

∴ *x* ≠ -3 ⇒ *x* = 7

So, the present age of Rehman = 7 years.

In a class test, the sum of Shefali's marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

SOLUTION:
**Soln. :** Let Shefali's marks in Mathematics = *x*

∴ Marks in English = (30 - *x*)

[∴ Sum of the marks in English and Mathematics = 30]

Now, according to the condition

(*x* + 2) × [(30 - *x*) - 3] = 210

⇒ (*x* + 2) × (30 - *x* - 3) = 210

⇒ (*x* + 2)(- *x* + 27) = 210

⇒ -*x*^{2} + 25*x* + 54 = 210

⇒ -*x*^{2} + 25*x* + 54 - 210 = 0

⇒ -*x*^{2} + 25*x* - 156 = 0

⇒ *x*^{2} - 25*x* + 156 = 0 …(1)

Now, comparing (1) with *ax*^{2} + *bx* + *c* = 0, we have *a* = 1, *b* = -25, *c* = 156

∴ *b*^{2} - 4*ac* = (-25)^{2} - 4(1)(156) = 625 - 624 = 1

Since,

⇒

Taking positive sign,

Taking negative sign,

When *x* = 13, then 30 - 13 = 17

When *x* = 12, then 30 - 12 = 18

Thus, marks in Mathematics = 13, marks in English = 17

marks in Mathematics = 12,

marks in English = 18.

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

SOLUTION:
**Soln. :** Let the shorter side (*i.e*., breadth) = *x* metres

∴ The longer side (length) = (*x* + 30) metres

In a rectangle,

⇒

⇒

⇒ (*x* + 60)^{2} = 2*x*^{2} + 60*x* + 900

⇒ *x*^{2} + 120*x* + 3600 = 2*x*^{2} + 60*x* + 900

⇒ 2*x*^{2} - *x*^{2} + 60*x* - 120*x* + 900 - 3600 = 0

⇒ *x*^{2} - 60*x* - 2700 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0, we have *a* = 1, *b* = -60, *c* = -2700

∴ *b*^{2} - 4*ac* = (-60)^{2} - 4(1)(-2700)

⇒ *b*^{2} - 4*ac* = 3600 + 10800

⇒ *b*^{2} - 4*ac* = 14400

Since,

⇒

Taking +ve sign,

Taking -ve sign,

Since breadth cannot be negative,

so, *x* ≠ -30 ⇒ *x* = 90

∴ *x* + 30 = 90 + 30 = 120

Thus, the shorter side = 90 m. The longer side = 120 m.

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

SOLUTION:
**Soln. :** Let the larger number be *x*.

Since, (smaller number)^{2} = 8(larger number)

⇒ (smaller number)^{2} = 8x

⇒ smaller number =

Now, according to the condition,

⇒ *x*^{2} - 8*x* = 180 ⇒ *x*^{2} - 8*x* - 180 = 0

Comparing (1) with *ax*^{2} + *bx* + *c* = 0, we have *a* = 1, *b* = -8, *c* = -180

∴ *b*^{2} - 4ac = (-8)^{2} - 4(1)(-180) = 64 + 720 = 784

Since,

∴ [∴ ]

Taking positive sign,

Taking negative sign,

But *x* = -10 is not admissible,

∴ The larger number = 18

⇒ Smaller number = = = ±12

Thus, the smaller number = 12 or -12

Thus, the two numbers are 18 and 12, 18 and -12.

A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Find the speed of the train.

SOLUTION:**Soln. :** Let the uniform speed of the train be *x* km/hr

Since, time taken by the train

⇒

When speed is 5 km/hr more, then time taken is 1 hour less.

⇒ ⇒

⇒ ⇒

⇒

⇒ - 5 × 360 = -1(*x*^{2} + 5*x*)

⇒ - 5 × 360 = -*x*^{2} - 5*x*

⇒ *x*^{2} + 5*x* - 1800 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0, we have *a* = 1, *b* = 5, *c* = -1800

∴ *b*^{2} - 4*ac* = (5)^{2} - 4(1)(-1800)

= 25 + 7200 = 7225

Since,

∴

Taking positive sign,

Taking negative sign,

Since, the speed of a vehicle cannot be negative,

So, *x* = - 45 (rejected)

Thus, *x* = 40 ⇒ speed of the train = 40 km/hr.

Two water taps together can fill a tank in hours. The tap of larger diameter takes

10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

**Soln. :** Let the smaller tap fills the tank in *x* hours

∴ The larger tap fills the tank in (*x* - 10) hours.

Amount of water flowing through both taps in one hour

Now, according to the condition,

⇒ ⇒

⇒ 8*x*^{2} - 80*x* = 150*x* - 750

⇒ 8*x*^{2} - 80*x* - 150*x* + 750 = 0

⇒ 8*x*^{2} - 230*x* + 750 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0, we get

*a* = 8, *b* = -230, *c* = 750

∴ *b*^{2} - 4*ac* = (-230)^{2} - 4(8)750

= 52900 - 24000 = 28900 > 0

Since,

⇒

[∴ ]

Taking positive sign,

Taking negative sign,

which is not possible.

[∴ Time cannot be negative]

∴ *x* = 25 ⇒ *x* - 10 = 25 - 10 = 15

Thus, time to fill the tank by the smaller tap alone = 25 hours and by larger tap alone = 15 hours.

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11 km/hr more than that of the passenger train, find the average speed of the two trains.

SOLUTION:
**Soln. :** Let the average speed of the passenger train be *x* km/hr

∴ Average speed of the express train

= (*x* + 11) km/hr

Total distance covered = 132 km

Time taken by the passenger train

Time taken by the express train

According to the condition,

we get

⇒ ⇒

⇒ ⇒

⇒ -11(132) = -1(*x*^{2} + 11*x*)

⇒ -1452 = -1(*x*^{2} + 11*x*)

⇒ *x*^{2} + 11*x* - 1452 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0,

we get *a* = 1, *b* = 11, *c* = -1452

∴ *b*^{2} - 4*ac* = (11)^{2} - 4(1)(-1452)

⇒ *b*^{2} - 4*ac* = 121 + 5808 = 5929

Since,

⇒

∴ Taking +ve sign,

Taking -ve sign,

But average speed cannot be negative so,
*x* ≠ -44 ∴ *x* = 33

⇒ Average speed of the passenger train

= 33 km/hr

∴ Average speed of the express train

= (*x* + 11) km/hr = (33 + 11) km/hr = 44 km/hr

Sum of the areas of two squares is 468 m^{2}. If the difference of their perimeters is 24 m, find the sides of the two squares.

**Soln. :** Let the side of the smaller square be *x* m

⇒ Perimeter of the smaller square = 4*x* m

∴ Perimeter of the larger square = (4*x* + 24) m

⇒ side of the larger square

∴ Area of the smaller square = (side)^{2} = (*x*)^{2} m^{2}

Area of the larger square = (*x* + 6)^{2} m^{2}

According to the condition, *x*^{2} + (*x* + 6)^{2} = 468

⇒ *x*^{2} + *x*^{2} + 12*x* + 36 = 468

⇒ 2*x*^{2} + 12*x* - 432 = 0

⇒ *x*^{2} + 6*x* - 216 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0, we get

∴ *a* = 1, *b* = 6, *c* = -216

∴ *b*^{2} - 4*ac* = (6)^{2} - 4(1)(-216) = 36 + 864 = 900

Since,

∴

∴ Taking positive sign, we have

Taking negative sign, we get

But the length of a square cannot be negative

∴ *x* = 12

⇒ length of the smaller square = 12 m

Thus, the length of the larger square

= *x* + 6 = 12 + 6 = 18 m

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them :

(i) 2*x*^{2} - 3*x* + 5 = 0

(ii)

(iii) 2*x*^{2} - 6*x* + 3 = 0

**Soln. :** (i) Comparing the given quadratic equation with *ax*^{2} + *bx* + *c* = 0, we have
*a* = 2, *b* = -3, *c* = 5

∴ Discriminant = *b*^{2} - 4*ac* = (-3)^{2} - 4(2)(5)

= 9 - 40 = - 31 < 0

Since *b*^{2} - 4*ac* is negative

∴ The given quadratic equation has no real roots.

(ii) Comparing the given quadratic equation with *ax*^{2} + *bx* + *c* = 0, we get
*a* = 3, *b* = , *c* = 4

∴ = (16 × 3) - 48

= 48 - 48 = 0

Thus, the given quadratic equation has two real roots which are equal. Hence, the roots are

⇒

⇒

Thus,

(iii) Comparing it with the general quadratic equation, we have

*a* = 2, *b* = -6, *c* = 3

∴ *b*^{2} - 4*ac* = (-6)^{2} - 4(2)(3)

= 36 - 24 = 12 > 0

∴ The given quadratic equation has two real and distinct roots, which are given by

⇒

Thus, the roots are =

Find the values of *k* for each of the following quadratic equations so that they have two equal roots

(i) 2*x*^{2} + *kx* + 3 = 0

(ii) *kx*(*x* - 2) + 6 = 0

**Soln. :** (i) Comparing the given quadratic equation with *ax*^{2} + *bx* + *c* = 0, we get
*a* = 2, *b* = *k*, *c* = 3

∴ *b*^{2} - 4*ac* = (*k*)^{2} - 4(2)(3) = *k*^{2} - 24

∴ For a quadratic equation to have equal roots, *b*^{2} - 4*ac* = 0

∴ ⇒

Thus, the required values of *k* are

(ii) Comparing *kx*(*x* - 2) + 6 = 0
*i.e*., *kx*^{2} - 2*kx* + 6 = 0 with *ax*^{2} + *bx* + *c* = 0, we get

*a* = *k*, *b* = -2*k*, *c* = 6

∴ *b*^{2} - 4*ac* = (-2*k*)^{2} - 4(*k*)(6) = 4*k*^{2} - 24*k*

Since, the roots are real and equal,

∴ *b*^{2} - 4*ac* = 0 ⇒ 4*k*^{2} - 24*k* = 0

⇒ 4*k*(*k* - 6) = 0

⇒ 4*k* = 0 or *k* - 6 = 0

⇒ *k* = 0 or *k* = 6

But *k* cannot be 0, otherwise, the given equation is no more quadratic. Thus, the required value of *k* = 6.

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 sq. metre ? If so, find its length and breadth.

SOLUTION:
**Soln. :** Let the breadth be *x* metres

∴ Length = 2*x* metres

Now, Area = Length × Breadth

= 2*x* × *x* metres^{2} = 2*x*^{2} sq.metre.

According to the given condition, 2*x*^{2} = 800

Therefore, *x* = 20 and *x* = -20

But *x* = -20 is not possible

[∴ breadth cannot be negative]

∴ *x* = 20 ⇒ 2*x* = 2 × 20 = 40

Thus, length = 40 m and breadth = 20 m

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

SOLUTION:
**Soln. :** Let the age of one friend = *x* years

∴ The age of the other friend = (20 - *x*) years

[∴ Sum of their ages is 20 years]

Four years ago

Age of one friend = (*x* - 4) years

Age of other friend = (20 - *x* - 4) years

= (16 - *x*) years

According to the condition, (*x* - 4) × (16 - *x*) = 48

⇒ 16*x* - 64 - *x*^{2} + 4*x* = 48

⇒ - *x*^{2} + 20*x* - 64 - 48 = 0

⇒ - *x*^{2} + 20*x* - 112 = 0

⇒ *x*^{2} - 20*x* + 112 = 0 …(1)

Here, *a* = 1, *b* = - 20 and *c* = 112

∴ *b*^{2} - 4*ac* = (-20)^{2} - 4(1)(112)

= 400 - 448 = -48 < 0

Since *b*^{2} - 4*ac* is less than 0.

∴ The quadratic equation (1) has no real roots.

Thus, the given equation is not possible.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m^{2}? If so, find its length and breadth.

**Soln. :** Let the breadth of the rectangle be *x* m.

Since, the perimeter of the rectangle = 80 m.

∴ 2[Length + breadth] = 80

2[Length + *x*] = 80

⇒

⇒ Length = (40 - *x*) m

∴ Area = (40 - *x*) × *x* sq.m = 40*x* - *x*^{2}

Now, according to the given condition,

Area of the rectangle = 400 m^{2}

∴ 40*x* - *x*^{2} = 400

⇒ - *x*^{2} + 40*x* - 400 = 0

⇒ *x*^{2} - 40*x* + 400 = 0 …(1)

Comparing (1) with *ax*^{2} + *bx* + *c* = 0, we get
* a* = 1, *b* = -40, *c* = 400

∴ *b*^{2} - 4*ac* = (-40)^{2} - 4(1)(400) = 1600 - 1600 = 0

Thus, the equation (1) has two equal and real roots.

∴ Breadth, *x* = 20 m

∴ Length = (40 - *x*) = (40 - 20) m = 20 m.

Since Length = Breadth ∴ This rectangle is a square.