When we equate quadratic polynomial of the form ax2 + bx + c, to zero, we get a quadratic equation.
A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. Any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, is a quadratic equation. ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.
A real number α is called a root of the quadratic equation
ax2 + bx + c = 0, a ≠ 0 if aα2 + bα + c = 0.
A quadratic polynomial can have at most two zeroes. So, any quadratic equation can have atmost two roots.
The roots of ax2 + bx + c = 0 are
[-b+√(b2 – 4ac)]/2a and [(-b-√(b2 – 4ac)]/2a, if b2 – 4ac ＞= 0.
If b2– 4ac ＜ 0, the equation will have no real roots.
1. 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
2. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 m2. 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) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
1. Find the roots of the following quadratic equations by factorisation:
(i) x2 – 3x – 10 = 0 (ii) 2x2 + x – 6 = 0
(iii) (√2) x2 + 7 x + 5 (√2) = 0 (iv) 2x2 – x +(1/8)= 0
(v) 100 x2 – 20x + 1 = 0
2. Solve the problems given in Example 1.
(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 Rs 750. We would like to find out the number of toys produced on that day.
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 Rs. 90, find the number of articles produced and the cost of each article.
1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i) 2x2 – 7x + 3 = 0 (ii) 2x2 + x – 4 = 0 (iii) 4x2 + 4 (√3)x + 3 = 0 (iv) 2x2 + x + 4 = 0
2. Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
EXERCISE 4.3 Q3-6 Solved
3. Find the roots of the following equations:
(i) x – (1/x) = 3, x ≠ – 4, 7
(ii) [1/(x+4) – 1/(x-7)] = 11/30 , x ≠ – 4, 7
4. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.
5. In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.
6. The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.
EXERCISE 4.3 Q7-11 Solved
7. The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.
8. A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.
9. Two water taps together can fill a tank in 9 3/8 hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.
10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.
11. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of the two squares.
Exercise 4.4 Q1-5 Solved
1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x^2 – 3x + 5 = 0
(ii) 3x^2 – 4√3 x + 4 = 0
(iii) 2x^2 – 6x + 3 = 0
2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x^2 + 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.