Become A Master Of Quadratic Equations.

Representing word problems as Quadratic Equations

Represent the following situations mathematically:

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.

Represent the following situations mathematically:

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.

The hypotenuse of a right angled triangle is 6 metres more than twice the shortest side. If the third side is 2 meters less than the hypotenuse, find the sides of the triangle.

A man travels a distance of 300 km at a uniform speed. If the speed of the train is increased by 5 km an hour, the journey would have taken two hours less. Find the original speed of the train.

Method to check whether a given equation is quadratic or not & method to determine unknown constant involved in a Q.E.

Check whether the following are quadratic equations:

(i) (x – 2)2 + 1 = 2x – 3

(ii) x(x + 1) + 8 = (x + 2) (x – 2)

(iii) x (2x + 3) = x2 + 1

(iv) (x + 2)3 = x3 – 4

Which one of the following is not a quadratic equation?

(A) (x + 2)2 = 2(x + 3)       

(B)  x2 + 3x = (–1)(1 – 3x)2

(C) (x + 2) (x – 1) = x2 – 2x – 3

(D)  x3 – x2 + 2x + 1 = (x + 1)3

Check whether 16/x – 1 = 15/(x+1) is a quadratic equation?

Check whether 1/(x+1) + 2/(x+2) = 4/(x+4) is a quadratic equation?

Learning Objectives:

ü How to find Solutions or Roots of a Quadratic Equation.

ü Different methods to find Solution of a Quadratic Equation.

Find the roots of the equation 2x2 – 5x + 3 = 0, by factorization.

Find the roots of the quadratic equation 6x2 – x – 2 = 0.

Find the roots of the quadratic equation 3x2 − 2 x + 2 = 0.

A charity trust decides to build a prayer hall having a carpet area of 300 square metres with its length one metre more than twice its breadth. What should be the length and breadth of the hall?

Solution of a Quadratic Equation by Completing the Square method:

Find the roots of the equation 2x^2 – 5x + 3 = 0,

by the method of completing the square.

Find the roots of the equation 5x^2 – 6x – 2 = 0

by the method of completing the square.

Find the roots of x^2 – 4x – 8 = 0 by the

method of completing square.

Find the roots of 4x2 + 3x + 5 = 0 by the

method of completing the square.

Solution of a Quadratic Equation by Quadratic Formula:

Find the roots of the following quadratic equations,

if they exist, using the quadratic formula:

(i) 3x2 – 5x + 2 = 0

(ii) x2 + 4x + 5 = 0

(iii) 2x2 – 2√2x + 1 = 0

Find the roots of the following equations:

(i) x + 1/x = 3,  x ≠ 0

(ii) 1/x - 1/(x-2) = 3,  x ≠ 0,

The area of rectangular plot is 528m2. 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. Solve the situation by using the quadratic formula.

Solution of a Quadratic Equation - word problems

Find two consecutive odd positive integers,

sum of whose squares is 290.

A rectangular park is to be designed whose breadth is 3 m less than its length. Its area is to be 4 square meters more than the area of a park that has already been made in the shape of an isosceles triangle with its base as the breadth of the rectangular park and of altitude 12 m (see figure). Find its length and breadth.

A motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

P & Q are centres of circles of radii 9 cm and 2 cm respectively. PQ = 17 cm. R is the centre of the circle of radius x cm which touches given circles externally. Given that angle PRQ is 90°. Write an equation in x and solve it.

A pole has to be erected at a point on the boundary of a circular park of diameter 13 meters in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 meters. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Nature of Roots :

Find the discriminant of the quadratic equation

2x2 – 4x + 3 = 0, and hence find the nature of its roots.

Find the discriminant of the equation 3x2 – 2x + = 0 and

hence find the nature of its roots. Find them, if they are real.

If – 3 is a root of quadratic equation 2x2 + px – 15 = 0, while the quadratic equation x2 − 4px + k = 0 has equal roots , find the value of ‘ k ’ .

If the equation (1 + m2) x2 + 2 mcx + (c2 − a2) = 0

has equal roots, prove that c2 = a2 (1 + m2).

Find the value of ‘k ’ for the quadratic equation

kx (x − 2) + 6 = 0, so that they have two equal roots.

Find the non-zero values of ‘ k ’ for which the roots of the quadratic equation 9x2 - 3kx + k = 0 are real and equal.

Check whether the following are Quadratic Equations.

(i) (x + 1)2= 2 (x − 3)

(ii) x2 - 2x = (−2) (3 − x)

(iii) (x − 2) (x + 1) = (x − 1) (x + 3)

(iv) (x − 3) (2x + 1) = x (x + 5)

(v) (2x − 1) (x − 3) = (x + 5) (x − 1)

(vi) x2 + 3x +1 = (x - 2)2

(vii) (x + 2)3 = 2x(x2 - 1)

(viii)  x3 - 4x2 - x + 1 = (x - 2)3

Represent the following situations in the form of Quadratic Equations:

(i) The area of rectangular plot is 528m2. 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) after 3 years will be 360. We would like to find Rohan's present age.

(iv) A train travels a distance of 480 km at uniform speed. If, the speed had been 8km/h less, then it would have taken 3 hours more to cover the same distance. We need to find speed of the train.

Find the roots of the following

quadratic equations by factorization:

(i)  x2 – 3x – 10 = 0   

(ii)  2x2 + x – 6 = 0   

(iii) √2x2 + 7x + 5√2 = 0

(iv) 2x2 - x + 1/8 = 0   

(v) 100x2 - 20x + 1 = 0 

Represent the following situations mathematically:

(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.

Find two numbers whose sum is 27

and product is 182.

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

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

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that 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.