Class 10 (All in one) – Quadratic Equations

If (x + a) is a factor of 2x² + 2ax + 5x + 10 = 0, then find the value of a.

Solve the equation by factorization.
t² + 3t – 10 = 0

Solve the equation by factorization.
21x² – 2x + $\frac{1}{21}$ = 0

Solve the quadratic equation
$3 x^{2} + 2 \sqrt{5} x - 5 = 0$

Solve the quadratic equation
$x^{2} - (\sqrt{2} + 1) x + \sqrt{2} = 0$

Solve the quadratic equation
$\sqrt{3} x^{2} + 11 x + 6 \sqrt{3} = 0$

Solve the quadratic equation by method of factorization.
$5 x + \frac{1}{x} = 6; x \neq 0$

Solve the quadratic equation by method of factorization.
$\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}; x \neq 3, - 5$

Solve the quadratic equation by method of factorization.
$\frac{21}{x^{2}} - \frac{29}{x} - 10 = 0; x \neq 0$

10.

Using factorization method, solve the quadratic equation.
$\frac{x + 3}{x + 2} = \frac{3 x - 7}{2 x - 3}$

11.

Using factorization method, solve the quadratic equation.
$2 (\frac{x + 2}{2 x - 3}) - 9 (\frac{2 x - 3}{x + 2}) = 3$

12.

Write the discriminant of the following
(i) 3x² – 4x + 7 = 0
(ii) $\sqrt{3} x^{2} - 2 \sqrt{2} x - 2 \sqrt{3} = 0$
(iii) 2x² + 3x + m = 0
(iv) x² + x + 7 = 0
(v) (4x – 3)² + 20x = 11
(vi) a²x² + 8abx + 4b² = 0

13.

Determine whether the roots of the quadratic equation are real or not.
$3 \sqrt{3} x^{2} + 10 x + \sqrt{3} = 0$

14.

Find the value of K for which following equation has real and equal roots
(i) 4x² + Kx + 9 = 0
(ii) x² – 2(K + 1)x + K² = 0
(iii) $K x^{2} - 2 \sqrt{5} x + 4 = 0$
(iv) $x^{2} - 2 x (1 + 3 K) + 7 (3 + 2 K) = 0$

15.

Determine the value(s) of m for which the equation x² + m(4x + m – 1) + 2 = 0 has real roots.

16.

For what value(s) of n, the equation (n + 3)x² – (5 – n)x + 1 = 0 has coincident roots?

17.

Find the roots of the following quadratic equation by applying the quadratic formula.
abx² + (b² – ac)x – bc = 0, a, b ≠ 0

18.

If two numbers differ by 2 and their product is 360, then find the numbers.

19.

If the product of two consecutive natural numbers is 210, then determine the numbers.

20.

The denominator of a fraction is 3 more than its numerator. The sum of the fraction and its reciprocal is $\frac{29}{10}$ . Find the fraction.

21.

The product of Ram’s age 5 year back and 9 year hence (in year) is 15. Find the present age of Ram.

22.

Divide 29 into two parts, so that the sum of the squares of the parts is 425.

23.

If the product of two consecutive odd numbers is 143, then find the numbers.

24.

Some students planned a picnic. The total budget for food was ₹2000. But 5 students failed to attend the picnic and thus the cost of food for each member increased by ₹20. How many students attended the picnic and how much did each student pay for the food?

25.

A person on tour has ₹360 for his expenses. If he extends his tour for 4 days, then he has to cut down his daily expenses by ₹3. Find the original duration of the tour.

26.

A rectangular field is 20 m long and 14 m wide. There is a path of equal width all around it, having an area of 111 sq. m. Find the width of the path.

27.

In a flight of 600 km, an aircraft was slowed down due to bad weather. Its average speed for the trip was reduced by 200 km/h and the time of flight increased by 30 min. Find the duration of the flight.

28.

The area of a right angled triangle is 480 cm². If the base of the triangle is 8 cm more than twice the height (altitude) of the triangle, then find the sides of the triangle.

29.

A two-digit number is such that the product of its digit is 35. When 18 is added to the number, the digits interchange their places. Find the number.

30.

Twenty seven years hence Sanjay’s age will be square of what it was 29 years ago. Find his present age.

31.

If b = 0, c < 0, is it true that the roots of x² + bx + c = 0 are numerically equal and opposite in sign? Justify your answer.

32.

If the discriminant of the equation 6x² – bx + 2 = 0 is 1, then find the value of b.

33.

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify your answer.

34.

Check whether the following statement is true or false. Justify your answer.
“Every quadratic equation has atleast one real root.”

35.

If the product of two consecutive integers is 306, then write the quadratic representation of this situation.

36.

If a number is added to twice its square, then the resultant is 21. Write the quadratic representation of this situation.

37.

Which constant must be added and subtracted to solve the quadratic equation $9 x^{2} + \frac{3}{4} x - \sqrt{2} = 0$ by the method of completing the square?

38.

A quadratic equation with integral coefficients has integral roots. Justify your answer.

39.

Does there exist a quadratic equation whose coefficients are all distinct irrationals but both the roots are rationals? Why?

40.

Solve for $x : \sqrt{6 x + 7} - (2 x - 7) = 0$

41.

Find the roots of the equation ax² + a = a²x + x

42.

Show that (x² + l)² – x² = 0 has no real roots.

43.

Find the roots of the quadratic equation a²b²x² + b²x – a²x – 1 = 0

44.

Solve for x.
$\sqrt{2 x + 9} + x = 13$

45.

Find the numerical difference of the roots of equation x² – 7x – 18 = 0

46.

If $\frac{1}{2}$ is a root of the equation $x^{2} + k x - \frac{5}{4} = 0,$ then find the value of k.

47.

In a cricket match. Harbhajan took three wickets less than twice the number of wickets taken by Zaheer. The product of the numbers of wickets taken by these two is 20. Represent the above situation in the form of a quadratic equation.

48.

Find the quadratic equation, if $x = \sqrt{5 + \sqrt{5 + \sqrt{5 + ... \infty}}}$ and x is a natural number.

49.

Solve for $x : \frac{16}{x} - 1 = \frac{15}{x + 1}; x \neq 0, - 1$

50.

Find the value of p, when $p x^{2} + (\sqrt{3} - \sqrt{2}) x - 1 = 0 and x = \frac{1}{\sqrt{3}}$ is one root of this equation.

51.

The difference of two numbers is 4. If the difference of their reciprocals is $\frac{4}{21},$ then find the two numbers.

52.

The sum of two number is 11 and the sum of their reciprocals is $\frac{11}{28} .$ Find the numbers.

53.

At t min past 2 pm, the time needed by the minute hand of a clock to show 3 pm was found to be 3 min less than $\frac{t^{2}}{4}$ min. Find t.

54.

Seven years ago, Varun's age was five times the square of Swati's age. Three years hence, Swati's age will be two-fifth of Varun's age. Find their present ages.

55.

Find the roots of the equation a²x² – 3abx + 2b² = 0 by the method of completing the square.

56.

The sum of a number and its positive square root is $\frac{6}{25} .$ Find the number.

57.

The numerator of a fraction is 3 less than its denominator. If 1 is added to the denominator, the fraction is decreased by $\frac{1}{15} .$
Find the fraction.

58.

Three consecutive natural numbers are such that the square of the middle number exceeds the difference of the squares of other two by 60. Find the numbers.

59.

Solve for $x : \frac{1}{a + b + x} = \frac{1}{a} + \frac{1}{b} + \frac{1}{x},$ where a, b, x ≠ 0 and a + b + x ≠ 0

60.

If the roots of the equation x² + 2cx + ab = 0 are real and unequal, then prove that the equation x² – 2(a + b)x + a² + b² + 2c² = 0 has no real roots.

61.

Solve for x:
$\frac{1}{(x - 1) (x - 2)} + \frac{1}{(x - 2) (x - 3)} = \frac{2}{3}; x \neq 1, 2, 3$

62.

Find the roots of 5⁽^x^{+ 1)} + 5^{(2 – x)} = 5³ + 1 by factorisation method.

63.

If –5 is a root of the quadratic equation 2x² + px – 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, then find the value of k.

64.

A piece of cloth costs ₹ 200. If the piece was 5 m longer and each metre of cloth costs ₹ 2 less, the cost of the piece would have remained unchanged. How long is the piece and what is the original rate per metre?

65.

₹ 6500 were divided equally among a certain number of persons. If there had been 15 more persons, each would have got ₹ 30 less. Find the original number of persons.

66.

A shopkeeper buys a number of books for ₹ 1200. If he had bought 10 more books for the same amount, each book would have cost him ₹ 20 less. How many books did he buy?

67.

If the equation (1 + m²)x² + (2mc)x + (c² – a²) = 0 has equal roots, then prove that c² = a² (1 + m²).

68.

A factory kept increasing its output by the same percentage every year. Find the percentage, if it is known that the output doubles in the last two years.

69.

Solve $x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}; x \neq 2.$

70.

If the roots of the equation (a – b)x² + (b – c)x + (c – a) = 0 are equal, then prove that 2a = b + c.

71.

There is a square field whose side is 44 m. A square flower bed is prepared in its centre leaving a gravel path all round the flower bed. The total cost of laying the flower bed and gravelling the path at ₹ 2.75 and ₹ 1.50 per m² respectively, is ₹ 4904. Find the width of gravel path.

72.

Due to some technical problems, an aeroplane started late by one hour from its starting point. The pilot decided to increase the speed of the aeroplane by 100 km/h from its usual speed to cover a journey of 1200 km in same time.
(i) Find the usual speed of the aeroplane.
(ii) What values (qualities) of the pilot are represented in the question?

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