#### Chapters

#### Balbharati Balbharati Class 10 Mathematics 1 Algebra

## Chapter 2: Quadratic Equations

#### Chapter 2: Quadratic Equations solutions [Page 34]

Write any two quadratic equations.

Decide which of the following are quadratic equations.

*x*^{2} + 5*x *– 2 = 0

Decide which of the following are quadratic equations.

*y*^{2 }= 5*y* – 10

Decide which of the following are quadratic equations.

\[y^2 + \frac{1}{y} = 2\]

Decide which of the following are quadratic equations.

\[x + \frac{1}{x} = - 2\]

Decide which of the following are quadratic equations.

(*m* + 2) (*m *– 5) = 0

Decide which of the following are quadratic equations.

*m*^{3 }+ 3*m*^{2} – 2 = 3*m*^{3 }

Write the following equations in the form *ax*^{2} + *bx* + *c*= 0, then write the values of *a*, *b*, *c* for each equation.

2*y* = 10 – *y*^{2 }

Write the following equations in the form *ax*^{2} + *bx* + *c*= 0, then write the values of *a*, *b*, *c* for each equation.

(*x* – 1)^{2} = 2*x* + 3

Write the following equations in the form *ax*^{2} + *bx* + *c*= 0, then write the values of *a*, *b*, *c* for each equation.

*x*^{2} + 5*x* = –(3 –* x*)

*ax*^{2} + *bx* + *c*= 0, then write the values of *a*, *b*, *c* for each equation.

3*m*^{2} = 2*m*^{2} – 9

*ax*^{2} + *bx* + *c*= 0, then write the values of *a*, *b*, *c* for each equation.

*x*^{2} – 9 = 13

Determine whether the values given against each of the quadratic equation are the roots of the equation.

*x*^{2} + 4*x* – 5 = 0 , *x* = 1, –1

Determine whether the values given against each of the quadratic equation are the roots of the equation.

2*m*^{2} – 5*m* = 0,

Find *k* if *x* = 3 is a root of equation *kx*^{2} – 10*x* + 3 = 0

One of the roots of equation 5*m*^{2} + 2*m* + *k *= 0 is \[\frac{- 7}{5}\] Complete the following activity to find the value of '*k*'.

#### Chapter 2: Quadratic Equations solutions [Pages 0 - 36]

Solve the following quadratic equations by factorisation.

*x*^{2} – 15*x* + 54 = 0

Solve the following quadratic equations by factorisation.

*x*^{2} + *x* – 20 = 0

Solve the following quadratic equations by factorisation.

2*y*^{2} + 27*y* + 13 = 0

Solve the following quadratic equations by factorisation.

5*m*^{2} = 22*m* + 15

Solve the following quadratic equations by factorisation.

2*x*^{2} – 2*x* +\[\frac{1}{2}\]=0

Solve the following quadratic equations by factorisation.

\[6x - \frac{2}{x} = 1\]

Solve the following quadratic equations by factorisation.

\[\sqrt{2} x^2 + 7x + 5\sqrt{2} = 0\] to solve this quadratic equation by factorisation, complete the following activity.

Solve the following quadratic equations by factorisation.

\[3 x^2 - 2\sqrt{6}x + 2 = 0\]

Solve the following quadratic equations by factorisation.

\[2m\left( m - 24 \right) = 50\]

Solve the following quadratic equations by factorisation.

\[25 m^2 = 9\]

Solve the following quadratic equations by factorisation.

\[7 m^2 = 21m\]

Solve the following quadratic equations by factorisation.

\[m^2 - 11 = 0\]

#### Chapter 2: Quadratic Equations solutions [Page 39]

Solve the following quadratic equations by completing the square method.

*x*^{2} + *x* – 20 = 0

Solve the following quadratic equations by completing the square method.

*x*^{2} + 2*x* – 5 = 0

Solve the following quadratic equations by completing the square method.

*m*^{2} – 5*m* = –3

Solve the following quadratic equations by completing the square method.

9*y*^{2} – 12*y* + 2 = 0

Solve the following quadratic equations by completing the square method.

2*y*^{2} + 9*y* +10 = 0

Solve the following quadratic equations by completing the square method.

5*x*^{2} = 4*x *+ 7

#### Chapter 2: Quadratic Equations solutions [Pages 43 - 44]

Compare the given quadratic equations to the general form and write values of *a*,*b*, *c*.

*x*^{2} – 7*x* + 5 = 0

Compare the given quadratic equations to the general form and write values of *a*,*b*, *c*.

2*m*^{2} = 5*m* – 5

Compare the given quadratic equations to the general form and write values of *a*,*b*, *c*.

*y*^{2} = 7*y*

Solve using formula.

*x*^{2} + 6*x* + 5 = 0

Solve using formula.

*x*^{2} – 3*x* – 2 = 0

Solve using formula.

3*m*^{2} + 2*m* – 7 = 0

Solve using formula.

5*m*^{2} – 4*m* – 2 = 0

Solve using formula.

\[y^2 + \frac{1}{3}y = 2\]

Solve using formula.

5*x*^{2} + 13*x* + 8 = 0

With the help of the flow chart given below solve the equation \[x^2 + 2\sqrt{3}x + 3 = 0\] using the formula.

#### Chapter 2: Quadratic Equations solutions [Pages 49 - 50]

Fill in the gaps and complete.

Fill in the gaps and complete.

Fill in the gaps and complete.

If α, β are roots of quadratic equation,

Find the value of discriminant.

*x*^{2} + 7*x* – 1 = 0

Find the value of discriminant.

2*y*^{2} – 5*y* + 10 = 0

Find the value of discriminant.

\[\sqrt{2} x^2 + 4x + 2\sqrt{2} = 0\]

Determine the nature of roots of the following quadratic equations.

*x*^{2} – 4*x* + 4 = 0

Determine the nature of roots of the following quadratic equations.

2*y*^{2 }– 7*y* +2 = 0

Determine the nature of roots of the following quadratic equations.

*m*^{2} + 2*m* + 9 = 0

Form the quadratic equation from the roots given below.

0 and 4

Form the quadratic equation from the roots given below.

3 and –10

Form the quadratic equation from the roots given below.

\[\frac{1}{2}, - \frac{1}{2}\]

Form the quadratic equation from the roots given below.

\[2 - \sqrt{5}, 2 + \sqrt{5}\]

Sum of the roots of a quadratic equation is double their product. Find *k* if equation *x*^{2} – 4k*x* + *k *+3 = 0

α, β are roots of *y*^{2} – 2*y* –7 = 0 find,

α^{2} + β^{2 }

α, β are roots of *y*^{2} – 2*y* –7 = 0 find,

α^{3} + β^{3 }

The roots of each of the following quadratic equations are real and equal, find k.

3*y*^{2 }+ k*y* +12 = 0

The roots of each of the following quadratic equations are real and equal, find k.

k*x* (*x* – 2) + 6 = 0

#### Chapter 2: Quadratic Equations solutions [Page 52]

Product of Pragati’s age 2 years ago and 3 years hence is 84. Find her present age.

The sum of squares of two consecutive natural numbers is 244; find the numbers.

In the orange garden of Mr. Madhusudan there are 150 orange trees. The number of trees in each row is 5 more than that in each column. Find the number of trees in each row and each column with the help of following flow chart.

Vivek is older than Kishor by 5 years. The sum of the reciprocals of their ages is \[\frac{1}{6}\] Find their present ages.

Suyash scored 10 marks more in second test than that in the first. 5 times the score of the second test is the same as square of the score in the first test. Find his score in the first test.

Mr. Kasam runs a small business of making earthen pots. He makes certain number of pots on daily basis. Production cost of each pot is Rs 40 more than 10 times total number of pots, he makes in one day. If production cost of all pots per day is Rs 600, find production cost of one pot and number of pots he makes per day.

Pratik takes 8 hours to travel 36 km down stream and return to the same spot. The speed of boat in still water is 12 km. per hour. Find the speed of water current.

Pintu takes 6 days more than those of Nishu to complete certain work. If they work together they finish it in 4 days. How many days would it take to complete the work if they work alone.

If 460 is divided by a natural number, quotient is 6 more than five times the divisor and remainder is 1. Find quotient and diviser.

In the adjoining fig. \[\square\] ABCD is a trapezium AB || CD and its area is 33 cm^{2} . From the information given in the figure find the lengths of all sides of the \[\square\]ABCD. Fill in the empty boxes to get the solution.

#### Chapter 2: Quadratic Equations solutions [Pages 0 - 54]

Choose the correct answers for the following questions.

(1) Which one is the quadratic equation ?

(A)\[\frac{5}{x} - 3 = x^2\] (B) x(x+5)=2

(C)n-1=2n (D)`1/x^2(x+2)=x`

Out of the following equations which one is not a quadratic equation ?

(A) `x^2+4x=11+x^2` (B) `x^2=4x `

(C)`5x^2=90+5` (D) `2x-x^2=x^2`

The roots of* x*^{2} +* kx* +* k* = 0 are real and equal, find *k. *

(A) 0 (B)4

(C)0 or 4 (4)2

For \[\sqrt{2} x^2 - 5x + \sqrt{2} = 0\] find the value of the discriminant.

(A)-5 (B)17

(C)`sqrt2` (D)`2sqrt2-5`

Which of the following quadratic equations has roots 3,5 ?

Out of the following equations, find the equation having the sum of its roots –5.

(A) `3x^2-15x+3=0` (B)`x^2-5x+3=0`

(C)`x^2+3x-5=0` (D)`3x^2+15x+3=0`

\[\sqrt{5} m^2 - \sqrt{5}m + \sqrt{5} = 0\] which of the following statement is true for this given equation ?

One of the roots of equation *x*^{2} + *mx* – 5 = 0 is 2; find *m.*

*(A) -2 (B)-1/2*

*(C)1/2 (D)2*

Which of the following equations is quadratic ?

\[x^2 + 2x + 11 = 0\]

Which of the following equations is quadratic ?

\[x^2 - 2x + 5 = x^2\]

Which of the following equations is quadratic ?

\[\left( x + 2 \right)^2 = 2 x^2\]

Find the value of discriminant for each of the following equations.

\[2 y^2 - y + 2 = 0\]

Find the value of discriminant for each of the following equations.

\[5 m^2 - m = 0\]

Find the value of discriminant for each of the following equations.

\[\sqrt{5} x^2 - x - \sqrt{5} = 0\]

One of the roots of quadratic equation \[2 x^2 + kx - 2 = 0\] is –2. find *k*.

Two roots of quadratic equations are given ; frame the equation.

10 and –10

Two roots of quadratic equations are given ; frame the equation.

\[1 - 3\sqrt{5} \text{ and } 1 + 3\sqrt{5}\]

Two roots of quadratic equations are given ; frame the equation.

0 and 7

Determine the nature of roots for each of the quadratic equation.

\[3 x^2 - 5x + 7 = 0\]

Determine the nature of roots for each of the quadratic equation.

\[\sqrt{3} x^2 + \sqrt{2}x - 2\sqrt{3} = 0\]

Determine the nature of roots for each of the quadratic equation.

\[m^2 - 4x - 3 = 0\]

Solve the following quadratic equations.

\[\frac{1}{x + 5} = \frac{1}{x^2}\]

Solve the following quadratic equations.

\[x^2 - \frac{3x}{10} - \frac{1}{10} = 0\]

Solve the following quadratic equations.

\[\left( 2x + 3 \right)^2 = 25\]

Solve the following quadratic equations.

\[m^2 + 5m + 5 = 0\]

Solve the following quadratic equations.

\[5 m^2 + 2m + 1 = 0\]

Solve the following quadratic equations.

\[x^2 - 4x - 3 = 0\]

Find m if (*m* – 12) *x*^{2} + 2(*m *–12) *x *+ 2 = 0 has real and equal roots.

The sum of two roots of a quadratic equation is 5 and sum of their cubes is 35, find the equation.

Find quadratic equation such that its roots are square of sum of the roots and square of difference of the roots of equation

\[2 x^2 + 2\left( p + q \right)x + p^2 + q^2 = 0\]

The difference between squares of two numbers is 120. The square of smaller number is twice the greater number. Find the numbers.

Ranjana wants to distribute 540 oranges among some students. If 30 students were more each would get 3 oranges less. Find the number of students.

Mr. Dinesh owns an agricultural farm at village Talvel. The length of the farm is 10 meter more than twice the breadth. In order to harvest rain water, he dug a square shaped pond inside the farm. The side of pond is \[\frac{1}{3}\] of the breadth of the farm. The area of the farm is 20 times the area of the pond. Find the length and breadth of the farm and of the pond.

A tank fills completely in 2 hours if both the taps are open. If only one of the taps is open at the given time, the smaller tap takes 3 hours more than the larger one to fill the tank. How much time does each tap take to fill the tank completely ?

## Chapter 2: Quadratic Equations

#### Balbharati Balbharati Class 10 Mathematics 1 Algebra

#### Textbook solutions for Class 10th Board Exam

## Balbharati solutions for Class 10th Board Exam Algebra chapter 2 - Quadratic Equations

Balbharati solutions for Class 10th Board Exam chapter 2 (Quadratic Equations) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Maharashtra State Board Balbharati Class 10 Mathematics 1 Algebra solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10th Board Exam Algebra chapter 2 Quadratic Equations are Quadratic Equations Examples and Solutions, Quadratic Equations, Roots of a Quadratic Equation, Nature of Roots, Relation Between Roots of the Equation and Coefficient of the Terms in the Equation Equations Reducible to Quadratic Form, Solutions of Quadratic Equations by Factorization, Solutions of Quadratic Equations by Completing the Square, Formula for Solving a Quadratic Equation.

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