Solve the following quadratic equation for x,

4x² + 4bx - (a² - b²) = 0

solve for x:

353.

Find the roots of the quadratic equation 

Find the value of k for which the equation x² + k(2x + k − 1) + 2 = 0 has real and equal roots. 

If the equation (1 + m²) x² + 2mcx + c² – a² = 0 has equal roots then show that c² = a² (1 + m²). 

Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. 

If x = 3 is one root of the quadratic equation x²-2kx -6 =0, the find the value of k.

A plane left 30 minutes late than its scheduled time and in order to reach the destination 1500 km away in time, it had to increase its speed by 100 km/h from the usual speed. Find its usual speed.

A motor boat whose speed is 18 km/hr in still water 1 hr more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.

A train travels at a certain average speed for a distance of 63 km and then travels at a distance of 72 km at an average speed of 6 km/hr more than its original speed.  It takes 3 hours to complete the total journey, what is the original average speed ?