ICSE solutions for Mathematics [English] Class 10 chapter 6 - Quadratic Equation [Latest edition]

Which of the following are quadratic equation:
(2x - 3) (x + 5) = 2 - 3x

Which of the following are quadratic equation:
`(x -(1)/x)^2` = 0.

Determine, if 3 is a root of the given equation
`sqrt(x^2 - 4x + 3) + sqrt(x^2 - 9) = sqrt(4x^2 - 14x + 16)`.

Examine whether the equation 5x² -6x + 7 = 2x² – 4x + 5 can be put in the form of a quadratic equation.

Find if x = – 1 is a root of the equation 2x² – 3x + 1 = 0.

(3x - 5)(2x + 7) = 0

48x² – 13x -1 = 0

`10x -(1)/x` = 3

`(2)/x^2 - (5)/x + 2` = 0

`sqrt(3)x^2 + 11x + 6sqrt(3)` = 0

`sqrt(3)x^2 + 10x + 7sqrt(3)` = 0

ax² + (4a² - 3b)x - 12 ab = 0

Without solving the following quadratic equation, find the value of ‘p’ for which the given equation has real and equal roots:
x² + (p – 3) x + p = 0

Find the value of k for which the following equation has equal roots:
(k − 12)x² + 2(k − 12)x + 2 = 0.

If one root of the equation 2x² – px + 4 = 0 is 2, find the other root. Also find the value of p.

Solve x^2/3 + x^1/3 - 2 = 0.

The sum of two natural numbers is 15 and the sum of their reciprocals is `3/10`. Find the numbers.

Find two consecutive natural numbers whose squares have the sum 221.

The sum of the squares of three consecutive natural numbers is 110. Determine the numbers.

If an integer is added to its square the sum is 90. Find the integer with the help of a quadratic equation.

Find two consecutive positive even integers whose squares have the sum 340.

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

In a two digit number, the unit’s digit is twice the ten’s digit. If 27 is added to the number, the digit interchange their places. Find the number.

A two digit positive number is such that the product of its digits is 6. If 9 is added to the number, the digits interchange their places. Find the number.

Five years ago, a woman’s age was the square of her son’s age. Ten years later her age will be twice that of her son’s age. Find:
The age of the son five years ago.

Five years ago, a woman’s age was the square of her son’s age. Ten years later her age will be twice that of her son’s age. Find:
The present age of the woman.

The length of verandah is 3m more than its breadth. The numerical value of its area is equal to the numerical value of its perimeter.
(i) Taking x, breadth of the verandah write an equation in ‘x’ that represents the above statement.
(ii) Solve the equation obtained in above and hence find the dimension of verandah.

A two digit number is such that the product of its digit is 14. When 45 is added to the number, then the digit interchange their places. Find the number.

In each of the following determine the; value of k for which the given value is a solution of the equation:
kx² + 2x - 3 = 0; x = 2

In each of the following determine the; value of k for which the given value is a solution of the equation:
3x² + 2kx - 3 = 0; x = `-(1)/(2)`

In each of the following determine the; value of k for which the given value is a solution of the equation:
x² + 2ax - k = 0; x = - a.

If x = 2 and x = 3 are roots of the equation 3x² – 2kx + 2m = 0. Find the values of k and m.

Solve the following equation and give your answer up to two decimal places:
x² - 5x - 10 = 0

Determine whether the given values of x is the solution of the given quadratic equation below:
6x² - x - 2 = 0; x = `(2)/(3), -1`.

Find whether the value x = `(1)/(a^2)` and x = `(1)/(b^2)` are the solution of the equation:
a²b²x² - (a² + b²) x + 1 = 0, a ≠ 0, b ≠ 0.

Solve using the quadratic formula x² – 4x + 1 = 0

Solve the quadratic equation:
`4sqrt(5)x^2 + 7x - 3sqrt(5) = 0`.

3a²x² + 8abx + 4b² = 0

`(x  - a/b)^2 = a^2/b^2`

Solve the equation 2x `-(1)/x` = 7. Write your answer correct to two decimal places.

Form the quadratic equation whose roots are:
`sqrt(3) and 3sqrt(3)`

Form the quadratic equation whose roots are:
`2 + sqrt(5) and 2 - sqrt(5)`.

Find the value of k for which the given equation has real roots:
kx² - 6x - 2 = 0

Find the value of k for which the given equation has real roots:
9x² + 3kx + 4 = 0.

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
3x² + 2x - 1 = 0

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
`2sqrt(3)x^2 - 2sqrt(2)x - sqrt(3) = 0`

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
9a²b²x² - 48abc + 64c²d² = 0, a ≠ 0, b ≠ 0

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x² - 5x + 7 = 0

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x² - 4x + 1 = 0

Without actually determining the roots comment upon the nature of the roots of each of the following equations:
x² + 5x + 15 = 0.

Solve the equation 3x² – x – 7 = 0 and give your answer correct to two decimal places.

Solve for x using the quadratic formula. Write your answer correct to two significant figures (x -1)² – 3x + 4 = 0.

Without solving the following quadratic equation, find the value of 'm' for which the given equation has real and equal roots.

x² + 2(m – 1)x + (m + 5) = 0

Solve the following by reducing them to quadratic equations:
x⁴ - 26x2 + 25 = 0

Solve the following by reducing them to quadratic equations:
z⁴ - 10z² + 9 = 0.

Solve for x : `9^(x + 2) -6.3^(x + 1) + 1 = 0`.

Solve for x: (x² - 5x)² - 7(x² - 5x) + 6 = 0; x ∈ R.

Solve the following equation by reducing it to quadratic equation:
`sqrt(3x^2 - 2) + 1 = 2x`.

Solve: (x + 2) (x - 5) (x - 6) (x + 1) = 144.

A two digit number is such that the product of the digits is 12. When 36 is added to this number the digits interchange their places. Determine the number.

The side (in cm) of a triangle containing the right angle are 5x and 3x – 1. If the area of the triangle is 60 cm². Find the sides of the triangle.

Rs. 480 is divided equally among ‘x’ children. If the number of children were 20 more then each would have got Rs. 12 less. Find ‘x’.

By increasing the speed of a car by 10 km/hr, the time of journey for a distance of 72 km. is reduced by 36 minutes. Find the original speed of the car.

A car covers a distance of 400 km at a certain speed. Had the speed been 12 km/hr more, the time taken for the journey would have been 1 hour 40 minutes less. Find the original speed of the car.

The speed of an express train is x km/hr arid the speed of an ordinary train is 12 km/hr less than that of the express train. If the ordinary train takes one hour longer than the express train to cover a distance of 240 km, find the speed of the express train.

Some students planned a picnic. The budget for the food was Rs. 480. As eight of them failed to join the party, the cost of the food for each member increased by Rs. 10. Find how many students went for the picnic.

Two pipes flowing together can fill a cistern in 6 minutes. If one pipe takes 5 minutes more than the other to fill the cistern, find the time in which each pipe would fill the cistern.

One fourth of a herd of camels was seen in the forest. Twice the square root of the herd had gone to mountains and the remaining 15 camels were seen on the bank of a river. Find the total number of camels.

An aeroplane travelled a distance of 400 km at an average speed of x km/hr. On the return journey the speed was increased by 40 km/hr. Write down the expression for the time taken for
The outward journey

An aeroplane travelled a distance of 400 km at an average speed of x km/hr. On the return journey the speed was increased by 40 km/hr. Write down the expression for the time taken for
the return Journey. If the return journey took 30 minutes less than the onward journey write down an equation in x and find its value.

Car A travels x km for every litre of petrol, while car B travels (x + 5) km for every litre of petrol.
Write down the number of litres of petrol used by car A and car B in covering a distance of 400 km.

Car A travels x km for every litre of petrol, while car B travels (x + 5) km for every litre of petrol.
If car A use 4 litre of petrol more than car B in covering the 400 km, write down and equation in x and solve it to determine the number of litre of petrol used by car B for the journey.

A shopkeeper purchases a certain number of books for Rs. 960. If the cost per book was Rs. 8 less, the number of books that could be purchased for Rs. 960 would be 4 more. Write an equation, taking the original cost of each book to be Rs. x, and Solve it to find the original cost of the books.

Two pipes running together can 1 fill a cistern in 11 1/9 minutes. If one pipe takes 5 minutes more than the other to fill the cistern find the time when each pipe would fill the cistern.

In each of the following find the values of k of which the given value is a solution of the given equation:
7x² + kx -3 = 0; x = `(2)/(3)`

In each of the following find the values of k of which the given value is a solution of the given equation:
x² - x(a + b) + k = 0, x = a

In each of the following find the values of k of which the given value is a solution of the given equation:
`"k"x^2 + sqrt(2)x- 4 = 0; x = sqrt(2)`

In each of the following find the values of k of which the given value is a solution of the given equation:
x² + 3ax + k = 0; x = a.

Solve the following quadratic equation by factorisation:
(x - 4) (x + 2) = 0

Solve the following quadratic equation by factorisation:
(2x + 3) (3x - 7) = 0

Solve the following quadratic equation by factorisation:
x² + 3x - 18 = 0

Solve the following quadratic equation by factorisation:
x² - 3x - 10 = 0

Solve the following quadratic equation by factorisation:
9x² - 3x - 2 = 0

Solve the following quadratic equation by factorisation:
2x² + ax - a² = 0 where a ∈ R.

Solve the following quadratic equation by factorisation method:
`x/(x + 1) + (x + 1)/x = (34)/(15') x ≠ 0, x ≠ -1`

Solve the following quadratic equation by factorisation method:
`(x + 3)/(x - 2) - (1 - x)/x = (17)/(4)`.

Solve the following quadratic equation:
`(1)/(a + b + x) = (1)/a + (1)/b + (1)/x, a + b ≠ 0`

Solve the following quadratic equation:
4x² - 4ax + (a² - b²) = 0 where a , b ∈ R.

Determine whether the given quadratic equations have equal roots and if so, find the roots:
x² + 5x + 5 = 0

Determine whether the given quadratic equations have equal roots and if so, find the roots:
x² + 2x + 4 = 0

Determine whether the given quadratic equations have equal roots and if so, find the roots:
`(4)/(3)x^2 - 2x + (3)/(4) = 0`

Determine whether the given quadratic equations have equal roots and if so, find the roots:
3x² - 6x + 5 = 0

Find the value of k so that sum of the roots of the quadratic equation is equal to the product of the roots:
kx² + 6x - 3k = 0, k ≠ 0

Find the value of k so that sum of the roots of the quadratic equation is equal to the product of the roots:
(k + 1)x² + (2k + 1)x - 9 = 0, k + 1 ≠ 0.

Find the values of k so that the sum of tire roots of the quadratic equation is equal to the product of the roots in each of the following:
kx² + 2x + 3k = 0

Find the values of k so that the sum of tire roots of the quadratic equation is equal to the product of the roots in each of the following:
2x² - (3k + 1)x - k + 7 = 0.

Solve the following by reducing them to quadratic equations:
`((7y - 1)/y)^2 - 3 ((7y - 1)/y) - 18 = 0, y ≠ 0`

Solve the following by reducing them to quadratic equations:
`sqrt(x/(1 -x)) + sqrt((1 - x)/x) = (13)/(6)`.

Solve (x² + 3x)² - (x² + 3x) -6 = 0.

Solve the following by reducing them to quadratic form:
`sqrt(y + 1) + sqrt(2y - 5) = 3, y ∈ "R".`

Solve the following by reducing them to quadratic form:
`sqrt(x^2 - 16) - (x - 4) = sqrt(x^2 - 5x + 4)`.

Solve: x(x + 1) (x + 3) (x + 4) = 180.

Solve the equation:
`6(x^2 + (1)/x^2) -25 (x - 1/x) + 12 = 0`.

Solve for x:
`(x + 1/x)^2 - (3)/(2)(x - 1/x)-4` = 0.

Solve the equation x⁴ + 2x³ - 13x² + 2x + 1 = 0.

Given that one root of the quadratic equation ax² + bx + c = 0 is three times the other, show that 3b² – 16ac.

If one root of the quadratic equation ax² + bx + c = 0 is double the other, prove that 2b² = 9 ac.

If the ratio of the roots of the equation
lx² + nx + n = 0 is p: q, Prove that
`sqrt(p/q) + sqrt(q/p) + sqrt(n/l) = 0.`

In each of the following determine whether the given values are solutions of the equation or not.
3x² - 2x - 1 = 0; x = 1

In each of the following determine whether the given values are solutions of the equation or not.
x² + 6x + 5 = 0;  x = -1, x = -5

In each of the following determine whether the given values are solutions of the equation or not
2x² - 6x + 3 = 0; x = `(1)/(2)`

In each of the following determine whether the given values are solutions of the equation or not.
6x² - x - 2 = 0; x = `-(1)/(2), x = (2)/(3)`

In each of the following determine whether the given values are solutions of the equation or not.
x² + `sqrt(2)` - 4 = 0; x = `sqrt(2)`, x = -2`sqrt(2)`

In each of the following determine whether the given values are solutions of the equation or not.
9x² - 3x - 2 = 0; x = `-(1)/(3), x = (2)/(3)`

In each of the following determine whether the given values are solutions of the equation or not.
x² + x + 1 = 0; x = 1, x = -1.

In each of the following, determine whether the given values are solution of the given equation or not:
x² - 3x + 2 = 0; x = 2, x = -1

In each of the following, determine whether the given values are solution of the given equation or not:
x² + x + 1 = 0; x = 0; x = 1

In each of the following, determine whether the given values are solution of the given equation or not:
x² - 3`sqrt(3)` + 6 = 0; x = `sqrt(3)`, x = -2`sqrt(3)`

In each of the following, determine whether the given values are solution of the given equation or not:
`x = 1/x = (13)/(6), x = (5)/(6), x = (4)/(3)`

In each of the following, determine whether the given values are solution of the given equation or not:
2x² - x + 9 = x² + 4x + 3; x = 2, x = 3

In each of the following, determine whether the given values are solution of the given equation or not:
`x^2 - sqrt(2) - 4 = 0; x = -sqrt(2), x = -2sqrt(2)`

In each of the following, determine whether the given values are solution of the given equation or not:
`a^2x^2 - 3abx + 2b^2 = 0; x = a/b, x = b/a`.