The product of two consecutive positive integers is 306. Form the quadratic equation to find the integers, if x denotes the smaller integer.

Let the first number = x
Then second number = x + 1
Their product = 306
x (x + 1) = 306
⇒ x^{2} + x – 306 = 0
Required quadratic equation will be x^{2} + x – 306 = 0

The height of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, form the quadratic equation to find the base of the triangle.

Let the base of a right triangle = x
Its height = x – 7
and hypotenuse = 13 cm
⇒ By Pythagoras Theorem
(Hypotenuse)^{2} = (Base)^{2} + (Height)^{2}
(13 )^{2} = x^{2} + (x – 7)^{2}
⇒ 169 = x^{2} + x^{2} – 14x + 49
⇒ 2x^{2} – 14x + 49 – 169 = 0
⇒ 2x^{2} – 14x – 120 = 0
⇒ x^{2} – 7x – 60 = 0 (Dividing by 2)
Hence required quadratic equation will be x^{2} – 7x – 60 = 0

A train travels 360 km at a uniform speed. If the speed had been 5 km/hr more, it would have taken 1 hour less for the same journey. Form the quadratic equation to find the speed of the train.

Total distance = 360 km
Let the uniform speed of the train = x km/hr
Time taken =

Find the roots of the following quadratic equation 2x² – 7x + 3 = 0 by the method of completing the square.

Find the value of k for which the quadratic equation (3k + 1) x² + 2(k + 1) x + 1 = 0 has equal roots. Also, find the roots.