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.