Check whether the following are quadratic equations:
(i) \((x + 1)^2 = 2(x - 3)\)
(ii) \(x^2 - 2x = (-2)(3 - x)\)
(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) \(x^2 + 3x + 1 = (x - 2)^2\)
(vii) \((x + 2)^3 = 2x(x^2 - 1)\)
(viii) \(x^3 - 4x^2 - x + 1 = (x - 2)^3\)
(i) Yes
(ii) Yes
(iii) No
(iv) Yes
(v) Yes
(vi) No
(vii) No
(viii) Yes
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 is one more than twice its breadth. Find the length and breadth.
(ii) The product of two consecutive positive integers is 306. 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. Find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed were 8 km/h less, the journey would take 3 hours more. Find the speed of the train.
(i) \(2x^2 + x - 528 = 0\), where \(x\) is the breadth (in metres).
(ii) \(x^2 + x - 306 = 0\), where \(x\) is the smaller integer.
(iii) \(x^2 + 32x - 273 = 0\), where \(x\) (in years) is Rohan’s present age.
(iv) \(u^2 - 8u - 1280 = 0\), where \(u\) (in km/h) is the speed of the train.
Find the roots of the following quadratic equations by factorisation:
(i) \(x^2 - 3x - 10 = 0\)
(ii) \(2x^2 + x - 6 = 0\)
(iii) \(\sqrt{2}x^2 + 7x + 5\sqrt{2} = 0\)
(iv) \(2x^2 - x + \dfrac{1}{8} = 0\)
(v) \(100x^2 - 20x + 1 = 0\)
(i) \(-2, 5\)
(ii) \(-2, \dfrac{3}{2}\)
(iii) \(-\dfrac{5}{\sqrt{2}}, -\sqrt{2}\)
(iv) \(\dfrac{1}{4}, \dfrac{1}{4}\)
(v) \(\dfrac{1}{10}, \dfrac{1}{10}\)
Solve the problems given in Example 1.
Example 1: Represent the following situations mathematically and solve them:
(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. How many marbles did each of them have originally?
(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. How many toys were produced on that day?
(i) \(9, 36\)
(ii) \(25, 30\)
Find two numbers whose sum is 27 and product is 182.
Numbers are 13 and 14.
Find two consecutive positive integers, the sum of whose squares is 365.
Positive integers are 13 and 14.
The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
5 cm and 12 cm
A cottage industry produces a certain number of pottery articles in a day. The cost of production of each article is 3 rupees more than twice the number of articles produced that day. If the total cost of production is ₹90, find the number of articles produced and the cost of each article.
Number of articles = 6, Cost of each article = ₹15
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) \(3x^2 - 4\sqrt{3}x + 4 = 0\)
(iii) \(2x^2 - 6x + 3 = 0\)
(i) Real roots do not exist.
(ii) Equal roots: \(\dfrac{2}{\sqrt{3}}, \dfrac{2}{\sqrt{3}}\)
(iii) Distinct roots: \(\dfrac{3 \pm \sqrt{3}}{2}\)
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) \(kx(x - 2) + 6 = 0\)
(i) \(k = \pm 2\sqrt{6}\)
(ii) \(k = 6\)
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.
Yes. Length = 40 m, Breadth = 20 m
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.
No
Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.
Yes. Length = 20 m, Breadth = 20 m