4. Quadratic Equations

4.1 Introduction

4.2 Quadratic Equations

Example 1:

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.

Example 2:

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

4.3 Solution of a Quadratic Equation by Factorisation

Example 3:

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

Example 4:

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

Example 5:

Find the roots of the quadratic equation 3x² - 2√6x + 2 = 0 .

Example 6:

Find the dimensions of the prayer hall discussed in Section 4.1.

Exercise 4.2

Find the roots of the following quadratic equations by factorisation:

(i) x² – 3x – 10 = 0 (ii) 2x² + x – 6 = 0

(iii) √2x² + 7x + 5√2 = 0 (iv) 2x² – x + 1/8 = 0

(v) 100 x² – 20x + 1 = 0

2. Solve the problems given in Example 1.

3. Find two numbers whose sum is 27 and product is 182.

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

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

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

4.4 Nature of Roots

Example 7:

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

Example 8:

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

Example 9:

Find the discriminant of the equation 3x² – 2x + 1/3 = 0 and hence find the nature of its roots. Find them, if they are real.

Exercise 4.3

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x² – 3x + 5 = 0 (ii) 3x² – 4√3x + 4 = 0

(iii) 2x2 – 6x + 3 = 0

2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.

(i) 2x2 + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0

3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m²? If so, find its length and breadth.

4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.