Quadratic Equations In One Variable
Class 10th Mathematics West Bengal Board Solution
- I write the quadratic polynomials from the following polynomials by understanding it.…
- Which of the following equations can be written in the form of ax^2 + bx + c = 0, where…
- Let us determine the power of the variable for which the equation x^6 - x^3 -2 = 0 will…
- Let us determine the value of ‘a’ for which the equation (a-2)x^2 + 3x + 5=0 will not…
- If x/4-x = 1/3x , (x not equal 0 , x not equal 4) be expressed in the form of ax^2 +…
- Let us express 3x^2 + 7x + 23 = (x + 4) (x + 3) = 2 in the form of the quadratic…
- Let us express the equation (x+2)^3 = x (x^2 - 1) in the form of ax^2 + bx + c = 0 (a…
- i. Divide 42 into two parts such that one part is equal to the square of the other…
- The length of the diagonal of a rectangular area is 15 m. and the length exceeds its…
- One person bought some kg. sugar in ₹ 80. If he would get 4 kg. more sugar with that…
- The distance between two stations is 300 km. A train went to second station from first…
- A clock seller sold a clock by purchasing it at ₹ 336. The amount of his profit…
- If the velocity of the stream is 2kms/hr, then the time taken by Ratanmajhi to cover 2…
- The time taken to clean out garden by Majid is 3 hours more than Mahim. Both of them…
- The unit digit of a two digit number exceeds it tens’ digit by 6 and the product of…
- There is a road of equal width around the outside of a rectangular playground having…
- x^2 + x+1 = 0,1 and -1 In each of the following cases, let us justify write whether…
- 8x^2 +7x=0,0 and -2 In each of the following cases, let us justify write whether the…
- x + 1/x = 13/6 , 5/6 and 4/3 In each of the following cases, let us justify write…
- x^2 - root 3x-6 = 0 , - root 3 and 2 root 3 In each of the following cases, let us…
- Let us calculate and write the value of k for which 2/3 will be a root the quadratic…
- Let us calculate and write the value of k for which -a will be a root the quadratic…
- If 2/3 and -3 are the two roots of the quadratic equation ax^2 + 7x + b = 0, then let…
- 3y^2 - 20 = 160 - 2y^2 Let us solve :
- (2x+1)^2 + (x+1)^2 = 6x + 47 Let us solve :
- (x-7)(x-9) = 195 Let us solve :
- 3x - 24/x = x/3 , x not equal 0 Let us solve :
- x/3 + 3/x = 15/x , x not equal 0 Let us solve :
- 10x - 1/x = 3 , x not equal 0 Let us solve :
- 2/x^2 - 5/x + 2 = 0 , x not equal 0 Let us solve :
- (x-2)/(x+2) + 6 (x+2/x-6) = 1 , x not equal -2 , 6 Let us solve :…
- 1/x-3 - 1/x+5 = 1/6 , x not equal 3 ,-5 Let us solve :
- x/x+1 + x+1/x = 2 1/12 , x not equal 0 ,-1 Let us solve :
- ax+b/a+bx = cx+d/c+dx [a not equal b , c not equal d] , x not equal - a/b , -c/d Let…
- (2x+1) + 3/2x+1 = 4 , x not equal - 1/2 Let us solve :
- x+1/2 + 2/x+1 = x+1/3 + 3/x+1 - 5/6 , x not equal -1 Let us solve :…
- 12x+17/3x+1 - 2x+15/x+7 = 3 1/5 , x not equal - 1/3 ,-7 Let us solve :…
- x+3/x-3 + 6 (x-3/x+3) = 5 , x not equal 3 ,-3 Let us solve :
- 1/a+b+x = 1/a + 1/b + 1/x , x not equal 0 , - (a+b) Let us solve :…
- (x+a/x-a)^2 - 5 (x+a/x-a) + 6 = 0 , x not equal a Let us solve :
- 1/x - 1/x+b = 1/a - 1/a+b , x not equal 0 ,-b Let us solve :
- 1/(x-1) (x-2) + 1/(x-2) (x-3) + 1/(x-3) (x-4) = 1/6 , x not equal 1 , 2 , 3 , 4 Let us…
- a/x-a + b/x-b = 2c/x-c , x not equal a , b , c Let us solve :
- x^2 - (root 3+2) x+2 root 3 = 0 Let us solve :
- The difference of two positive whole numbers is 3 and the sum of their squares is 117;…
- The base of a triangle is 18m. more than two times of its heights, if the area of the…
- If 5 times of a positive whole number is less by 3 than twice of its square, then let…
- The distance between two places is 200 km.; the time taken by motor car from one place…
- The area of the Amita’s rectangular land is 2000 sq.m. and perimeter of it is 180 m. By…
- The tens digit of a two digit number is less by 3 than the unit digit. If the product…
- There are two pipes in a water reservoir of our school. Two pipes together take 11 1/9…
- Porna and Pijush together complete a work in 4 days. If they work separately, then the…
- If the price of 1 dozen pen is reduced by ₹ 6, then 3 more pens will be got in ₹ 30.…
- Q10A1 The number of roots of a quadratic equation isA. one B. two C. three D. none of them…
- Q10A2 If in a ax^2 + bx + c quadratic equation, thenA. b ≠ 0 B. C ≠ 0 C. a ≠ 0 D. none of…
- Q10A3 The highest power of the variable in a quadratic equation isA. 1 B. 2 C. 3 D. none…
- Q10A4 The equation 4(5x^2 - 7x + 2)=5 (4x^2 - 6x + 3) isA. linear B. quadratic C. 3rd…
- Q10A5 The root/two roots of the equation x^2/x = 6 .A. 0 B. 6 C. 0 6 D. -6…
- Let us write the following statements whether they are true or false : i. (x-3)^2 =…
- Let us fill in the blank : i. If a = 0 and b ≠ 0 in the equation ax^2 + bx + c = 0,…
- Let us find the value of a, if one root of the equation x^2 +ax+3=0 is 1…
- Let us write the value of the other root if one root of the equation x^2 - (2 + b) x…
- Let us write the value of the other root if one root of the equation 2x^2 + kx + 4 =…
- Let us write the equation; if the difference of a proper fraction and its reciprocal…
- Let us write the values of a and b, if the two roots of the equation ax^2 + bx + 35 =…
Let Us Work Out 1.1
Question 1.I write the quadratic polynomials from the following polynomials by understanding it.
i. x2 – 7x + 2
ii. 7x5 – x(x + 2)
iii. 2x (x+5) + 1
iv. 2x – 1
Answer:
(i) The given equation is x2 – 7x + 2
This equation is of the form ax2 + bx + c = 0 where a,b,c are real numbers as a = 1 which is not equal to 0,b = -7 and c = 2.
Hence, this equation is a quadratic equation.
(ii) The given equation is 7x5 – x(x + 2)
This equation is not of the form ax2 + bx + c = 0 where a,b,c are real numbers as the highest degree of x in the equation is 5.
Hence, this equation is not a quadratic equation.
(iii) The given equation is 2x (x+5) + 1
Therefore, this equation is of the form ax2 + bx + c = 0 where a,b,c are real numbers as a = 2 which is not equal to 0,b = 10 and c = 1.
Also the highest degree of x in the equation is 2.
Hence, this equation is a quadratic equation.
(iv) The given equation is 2x-1
This is a linear equation with highest degree of x is 1.
This is of the form ax2 + bx + c = 0 where a,b,c are real numbers but here a = 0 which does not satisfy the condition of quadratic equations.
Hence, this equation is not a quadratic equation.
Question 2.
Which of the following equations can be written in the form of ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0, let us write it.
i. 
ii. 
iii. 
iv. (x - 2)2 = x2 – 4x + 4
Answer:
(i) The given equation is 
It can be written as 
Therefore, this equation is of the form 
where a,b,c are real numbers.
Here, a = 1 which is non-zero, b = -7 and c = 1
(ii) The given equation is 
It can be written as 
Therefore, this equation is not of the form where a,b,c are real numbers.
Here, the highest degree of x in the equation is 3.
So, it is not a quadratic equation
(iii) The given equation is 
This equation is not of the form 
where a,b,c are real numbers.
So, it is not a quadratic equation.
(iv) The given equation is 
It is not a quadratic equation as both the sides are same and not forming an equation.
Question 3.
Let us determine the power of the variable for which the equation x6 – x3 –2 = 0 will become a quadratic equation?
Answer:
The given expression is 
Let x3 = k
It can be written as 
We see that it is of the form where a,b,c are real numbers.
Therefore, the equation is quadratic for the power of variable = 3
Question 4.
Let us determine the value of ‘a’ for which the equation (a-2)x2 + 3x + 5=0 will not be a quadratic equation.
Answer:
The given equation is (a-2)x2 + 3x + 5=0
We can see that the equation is of the form 
where a,b and c are real numbers.
So, for the equation not to be quadratic, a’ = 0
Therefore, in the equation 
a’ = a-2
a’ = 0
a – 2 = 0
So, a = 2.
Therefore, the above equation will not be quadratic for a = 2.
Question 5.
If 
be expressed in the form of ax2 + bx + c = 0 (a ≠ 0), then let us determine the co-efficient of x.
Answer:
Given equation is 
It can be written as 
This equation is of the form 
Therefore, a = 2, b = 1 and c = -4
So, coefficient of x = b = 1
Question 6.
Let us express 3x2 + 7x + 23 = (x + 4) (x + 3) = 2 in the form of the quadratic equation ax2 + bx + c = 0 (a ≠ 0)
Answer:
The given equation is 
So, the equation is of the form
Here, a = 2, b = 0 and c = 9
Question 7.
Let us express the equation (x+2)3 = x (x2 - 1) in the form of ax2 + bx + c = 0 (a ≠ 0) and write the co-efficient of x2, x and x°.
Answer:
The given equation is 
So, the equation is of the form
Here, a = 6, b = 13 and c = 8
Coefficient of x2 = 6
Coefficient of x = 13
Coefficient of x0 = 8
Question 8.
Let us construct the quadratic equations in one variable from the following statements.
i. Divide 42 into two parts such that one part is equal to the square of the other part.
ii. The product two consecutive positive odd numbers is 143.
iii. The sum of the squares of two consecutive numbers is 313.
Answer:
(i) Let one part be x.
And the other part = x2
Therefore, according to equation
x2 + x = 42
⇒ x2 + x – 42 = 0
(ii) Let the positive odd number be x + 1
The other consecutive positive odd number = (x + 1) + 2 = x + 3
According to question,
(x + 1)(x + 3) = 143
⇒ x2 + 3x + x + 3 = 143
⇒ x2 + 4x + 3-143 = 0
⇒ x2 + 4x -140 = 0
(iii) Let the first number be x.
And the other consecutive number = x + 1
According to question,
x2 + (x + 1)2 = 313
⇒ x2 + x2 + 1 + 2x = 313
⇒ 2x2 + 2x + 1 – 313 = 0
⇒ 2x2 + 2x - 312 = 0
Question 9.
Let us construct the quadratic equations in one variable from the following statements.
The length of the diagonal of a rectangular area is 15 m. and the length exceeds its breadth by 3 m.
Answer:
Let the breadth of the rectangle be x m.
Therefore, the length = (x + 3) m
Given length of diagonal = 15 m
We know that the diagonal of rectangle = 
According to equation,
Therefore, the required quadratic equation is 
Question 10.
Let us construct the quadratic equations in one variable from the following statements.
One person bought some kg. sugar in ₹ 80. If he would get 4 kg. more sugar with that money, then the price of kg. sugar would be less by ₹ 1.
Answer:
We know that
price = quantity of sugar in kg X rate per kg
Let the quantity be x.
Price for x kg of sugar = Rs 80
therefore, 
According to question,
For the same price, Quantity has increased by 4kgs and rate has decreased by 1 .
As, price = quantity of sugar in kg × rate per kg
So, 
Hence the required quadratic equation is 
Question 11.
Let us construct the quadratic equations in one variable from the following statements.
The distance between two stations is 300 km. A train went to second station from first station with uniform velocity. If the velocity of the train is 5 km/hour more, then the time taken by the train to reach the second station would be lesser by 2 hours.
Answer:
As we know that ,
distance = speed × time
Let speed of train be v.
Therefore,
300 = v × time
According to question,
if velocity is increased by 5 then time is decreased by 2.
As, distance = speed × time
So, 
Hence the required quadratic equation is 
Question 12.
Let us construct the quadratic equations in one variable from the following statements.
A clock seller sold a clock by purchasing it at ₹ 336. The amount of his profit percentage is as much as the amount with which he bought the clock.
Answer:
Given Selling Price, SP = Rs 336
Let the Cost Price CP be x
According to question,
Hence the required quadratic equation is 
Question 13.
Let us construct the quadratic equations in one variable from the following statements.
If the velocity of the stream is 2kms/hr, then the time taken by Ratanmajhi to cover 2 kms in downstream and upstream is 10 hours.
Answer:
Let the velocity of boat be v.
Given , velocity of the stream = 2km/h
velocity in downstream = v + 2
velocity in upstream = v-2
total time = time taken during upstream + time taken during downstream
As, 
According to question,
Hence the required quadratic equation is 
Question 14.
Let us construct the quadratic equations in one variable from the following statements.
The time taken to clean out garden by Majid is 3 hours more than Mahim. Both of them together can complete the work in 2 hours.
Answer:
Let the time taken by mahim be x hours.
So, time taken by majib = x + 3 hours
Work completed by Mahim in 1 hr
Work completed by Majid in 1 hr
Both complete the work in 2 hrs .
So, 
Hence the required equation is 
Question 15.
Let us construct the quadratic equations in one variable from the following statements.
The unit digit of a two digit number exceeds it tens’ digit by 6 and the product of two digits is less by 12 from the number.
Answer:
Let the tens digit be x.
therefore, unit’s digit = x + 6
According to question,
Hence the required equation is 
Question 16.
Let us construct the quadratic equations in one variable from the following statements.
There is a road of equal width around the outside of a rectangular playground having the length 45 m. and breadth 40 m. and the area of the road is 450 sqm.
Answer:
Let the width of road be w m
area of road = area of rectangular paths + 4 squares at the corner.
So, 
Hence the required equation is 
Let Us Work Out 1.2
Question 1.In each of the following cases, let us justify & write whether the given values are the of the given quadratic equation:
x2 + x+1 = 0,1 and -1
Answer:
We know that the roots of the quadratic equation satisfies the equation.
So, if 1 and -1 are the roots of 
then they must satisfy the equation.
Therefore, putting 1 in the equation we have
= 12 + 1 + 1 = 1 + 1 + 1 = 3 which is not equal to 0.
So, it is not a root of the equation.
Putting -1 in the equation we have
= (-1)2 + (-1) + 1 = 1-1 + 1 = 1 which is not equal to 0.
So, it is also not a root of the equation.
Hence, 1 and -1 are not the roots of the equation.
Question 2.
In each of the following cases, let us justify & write whether the given values are the of the given quadratic equation:
8x2+7x=0,0 and -2
Answer:
We know that the roots of the quadratic equation satisfies the equation.
So, if 0 and -2 are the roots of 
then they must satisfy the equation.
Therefore, putting 0 in the equation we have
= 8 × 02 + 7 × 0 = 0 + 0 = 0
So, it is a root of the equation.
Putting -2 in the equation we have
= 8 × (-2)2 + 7 × (-2) = 32-14 = 18 which is not equal to 0.
So, it is also not a root of the equation.
Hence, 0 is the root of the equation.
Question 3.
In each of the following cases, let us justify & write whether the given values are the of the given quadratic equation:
and 4/3
Answer:
We know that the roots of the quadratic equation satisfies the equation.
So, if 
and 
are the roots of 
then they must satisfy the equation.
Therefore, putting in the equation we have
= 
= 
= 
which is not equal to 
.
So, it is not a root of the equation.
Putting in the equation we have
= 
= 
which is not equal to .
So, it is also not a root of the equation.
Hence, 
and are not the roots of the equation.
Question 4.
In each of the following cases, let us justify & write whether the given values are the of the given quadratic equation:
and 
Answer:
We know that the roots of the quadratic equation satisfy the equation.
So, if -√3 and 2√3 are the roots of 
then they must satisfy the equation.
Therefore, putting -√3 in the equation we have
= -√32-√3 × (-√3)-6 = 3 + 3-6 = 6-6 = 0
So, it is a root of the equation.
Putting 2√3 in the equation we have
= (2√3)2-√3 × (2√3)-6 = 12-6-6 = 12-12 = 0
So, it is also a root of the equation.
Hence, -√3 and 2√3 are the roots of the equation.
Question 5.
Let us calculate and write the value of k for which 2/3 will be a root the quadratic equation 7x2 + kx – 3 = 0.
Answer:
Given equation is 
It is given that 
is a root of the equation.
Therefore, it must satisfy the equation.
⇒
⇒
Hence , the value of k is 
.
Question 6.
Let us calculate and write the value of k for which –a will be a root the quadratic equation x2 + 3ax + k = 0.
Answer:
Given equation is 
It is given that -a is a root of the equation.
Therefore, it must satisfy the equation.
⇒
Hence, the value of k is 
.
Question 7.
If 2/3 and -3 are the two roots of the quadratic equation ax2 + 7x + b = 0, then let me calculate the values of a and b.
Answer:
Given equation is 
It is given that -3 and are the roots of the equation.
Therefore, they must satisfy the equation.
⇒
………eq (1)
And
⇒
⇒
………eq(2)
Now, putting the value of b in eq(2)
Therefore putting the value of a in eq(1).
Therefore, putting the value of a and b in the equation
Hence , the required equation is 
Question 8.
Let us solve :
3y2 – 20 = 160 – 2y2
Answer:
Hence , y = 6 or y = -6
Question 9.
Let us solve :
(2x+1)2 + (x+1)2 = 6x + 47
Answer:
Hence , x = 3 or x = -3
Question 10.
Let us solve :
(x-7)(x-9) = 195
Answer:
Hence, x = 22 or x = -6
Question 11.
Let us solve :
Answer:
Hence, x = 3 or x = -3
Question 12.
Let us solve :
Answer:
Hence, x = 6 or x = -6
Question 13.
Let us solve :
Answer:
Hence, x = 
or x = -
Question 14.
Let us solve :
Answer:
Hence, x = 
or x = 2
Question 15.
Let us solve :
Answer:
Question 16.
Let us solve :
Answer:
Question 17.
Let us solve :
Answer:
Question 18.
Let us solve :
Answer:
Question 19.
Let us solve :
Answer:
Let 2x + 1 = y
For y = 3,
2x + 1 = 3
⇒ 2x = 3-1
⇒ 2x = 2
⇒ x = 
= 1
For y = 1,
2x + 1 = 1
⇒ 2x = 1-1
⇒ 2x = 0
⇒ x = 0
So, x = 1 or x = 0
Question 20.
Let us solve :
Answer:
Let x + 1 = k
For, k = 1
x + 1 = 1
⇒ x = 1-1 = 0
For, k = -6
x + 1 = -6
⇒ x = -6-1 = -7
So, x = 1 or x = -7
Question 21.
Let us solve :
Answer:
⇒ 30x2 + 270x + 520 = 48x2 + 352x + 112
⇒ 18x2 + 82x – 408 = 0
⇒ 9x2 + 41x – 204 = 0
⇒ 9x2 + 68x – 27x – 204 = 0
⇒ x (9x + 68) – 3(9x + 68) = 0
⇒ (x – 3) (9x + 68) = 0
⇒ x = 3 and x = -68/3
Question 22.
Let us solve :
Answer:
Question 23.
Let us solve :
Answer:
Question 24.
Let us solve :
Answer:
Let 
For k = 3,
Therefore, x = 2a
For k = 2,
Therefore, x = 3a
So, x = 2a or x = 3a
Question 25.
Let us solve :
Answer:
Question 26.
Let us solve :
Answer:
We can write this equation as
Question 27.
Let us solve :
Answer:
Question 28.
Let us solve :
Answer:
Let Us Work Out 1.3
Question 1.The difference of two positive whole numbers is 3 and the sum of their squares is 117; by calculating, let us write the two numbers.
Answer:
Let one of the whole numbers be x.
Therefore, the other whole number is x + 3.
According to question,
As given the numbers are positive, so, the numbers are x = 6 and x + 3 = 6 + 3 = 9.
Question 2.
The base of a triangle is 18m. more than two times of its heights, if the area of the triangle is 360 sq.m., then let us determine the height of it.
Answer:
Let the height of the triangle be h.
Therefore, the base of triangle = 2h + 18.
According to question,
Area of triangle = 360
So, the height of triangle is 15m.
Question 3.
If 5 times of a positive whole number is less by 3 than twice of its square, then let us determine the number.
Answer:
Let the positive number be x.
According to question,
As the number is positive, so, x = 3.
Question 4.
The distance between two places is 200 km.; the time taken by motor car from one place to another is less by 2 hrs than the time taken by a zeep car. If the speed of the motor car is 5 km/hr. more than the speed of the zeep car, then by calculating let us write the speed of the motor car.
Answer:
Let time taken by motor car be t.
As, distance = speed × time
200 = speed of motor car × t
speed of motor car = 
Given, time taken by zeep = t + 2
Again, speed of zeep = 
According to question,
Speed of the motor car is 5 km/hr more than the speed of the zeep car.
So, 
So, speed of the motor car is 
= 25km/hr.
Question 5.
The area of the Amita’s rectangular land is 2000 sq.m. and perimeter of it is 180 m. By calculating, let us write the length and breadth of the Amita’s land.
Answer:
Let the length of rectangular land be l.
As, length × breadth = Area
breadth = 
.
Also given, perimeter = 180
According to question,
Hence, if the length = 40m then breadth = 50m
Or
If length = 50m then length = 40m
Question 6.
The tens digit of a two digit number is less by 3 than the unit digit. If the product of the two digits is subtracted from the number, the result is 15. Let us write the unit digit of the number by calculation.
Answer:
Let the unit digit be x
therefore, tens digit be x-3.
According to question,
Hence, the unit digit is 5 or 9.
Question 7.
There are two pipes in a water reservoir of our school. Two pipes together take 
minutes to fill the reservoir. If the two pipes are opened separately, then one pipe would take 5 minutes more time than the other pipe. Let us write by calculating the time taken to fill the reservoir separately by each of the pipes.
Answer:
Let time taken by one pipe to separately fill the reservoir be x.
therefore, time taken by another pipe be x + 5.
As, portion filled by one pipe in one min = 
portion filled by another pipe in one min = 
According to question,
So the time taken to fill the reservoir by first pipe is 20 hrs and second pipe is 25 hrs.
Question 8.
Porna and Pijush together complete a work in 4 days. If they work separately, then the time taken by Porna would be 6 days more than the time taken by Pijaush. Let us write, by calculating, the time taken by Porna alone to complete the work.
Answer:
Let time taken by pijush be x days.
Therefore, time taken by porna be x + 6 days.
According to question,
So, time taken by pijush to complete the work alone is 6 days , and by porna is 12 days.
Question 9.
If the price of 1 dozen pen is reduced by ₹ 6, then 3 more pens will be got in ₹ 30. Before the reduction of price, let us calculate the price of 1 donzen pen.
Answer:
Let the price of a dozen pen before reduction be r
As, price = pens × rate per dozen
30 = pens × r
So, number of pens before reduction is 
.
After reduction:
Again price = pens × rate per dozen
As rate can never be negative so, price of a dozen pen is
Rs. 11.30.
Question 10.
The number of roots of a quadratic equation is
A. one
B. two
C. three
D. none of them
Answer:
As we know that the highest degree of variable in quadratic equation is two so the number of roots is two.
Question 11.
If in a ax2 + bx + c quadratic equation, then
A. b ≠ 0
B. C ≠ 0
C. a ≠ 0
D. none of these
Answer:
In a quadratic equation the coefficient of x2 is non-zero.
Question 12.
The highest power of the variable in a quadratic equation is
A. 1
B. 2
C. 3
D. none of these
Answer:
As we know that the highest power of variable in quadratic equation is two.
Question 13.
The equation 4(5x2 - 7x + 2)=5 (4x2 – 6x + 3) is
A. linear
B. quadratic
C. 3rd degree
D. none of these
Answer:
Question 14.
The root/two roots of the equation 
.
A. 0
B. 6
C. 0 & 6
D. -6
Answer:
Question 15.
Let us write the following statements whether they are true or false :
i. (x-3)2 = x2 – 6x + 9 is a quadratic equation
ii. 5 is the only one root of the equation x2 = 25
Answer:
(i) False
As both sides are same, so it is not a quadratic equation.
(ii) False
Question 16.
Let us fill in the blank :
i. If a = 0 and b ≠ 0 in the equation ax2 + bx + c = 0, then the equation is a …………………. equation.
ii. The two roots of the equation x2 = 6x are ……………. & …………….. .
Answer:
(i) linear
if a = 0, the equation is bx + c = 0 which is linear.
(ii) 6 and 0
Question 17.
Let us find the value of a, if one root of the equation x2+ax+3=0 is 1
Answer:
The given equation is 
Given one of the roots is 1.
So, 1 must satisfy the equation.
∴ (1)2 + a × 1 + 3 = 0
⇒ 1 + a + 3 = 0
⇒ a + 4 = 0
⇒ a = -4
Therefore, the value of a is -4.
Question 18.
Let us write the value of the other root if one root of the equation x2 - (2 + b) x + 6 = 0 is 2.
Answer:
The given equation is 
Given one of the roots is 2.
So, 2 must satisfy the equation.
∴ (2)2-(2 + b) × 2 + 6 = 0
⇒ 4-4-2b + 6 = 0
⇒ -2b = -6
⇒ b = 
⇒ b = 3
Therefore, the value of b is 3.
∴ the equation is 
⇒
Therefore the other root is 3.
Question 19.
Let us write the value of the other root if one root of the equation 2x2 + kx + 4 = 0 is 2.
Answer:
The given equation is 
Given one of the roots is 2.
So, 2 must satisfy the equation.
∴ 2 × (2)2 + k × 2 + 4 = 0
⇒ 8 + 2k + 4 = 0
⇒ 2k + 12 = 0
⇒
Therefore, the value of k is -6.
∴ the equation is 
⇒
Therefore the other root is 1.
Question 20.
Let us write the equation; if the difference of a proper fraction and its reciprocal is 9/20.
Answer:
Let the proper fraction be x.
Its reciprocal is
According to question ,
Question 21.
Let us write the values of a and b, if the two roots of the equation ax2 + bx + 35 = 0 are -5 and -7
Answer:
The given equation is 
Given the roots are -5 and -7.
So, -5 and -7 must satisfy the equation.
∴ 2 × (2)2 + k × 2 + 4 = 0
⇒ 8 + 2k + 4 = 0
⇒ 2k + 12 = 0
⇒ 
Therefore, the value of k is -6.
∴ the equation is 
⇒ 
Therefore, the other root is -1.