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Quadratic Equations

Short Answer Type

351.

Solve the following quadratic equation for x,

4x² + 4bx - (a² - b²) = 0

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352.

solve for x:

$square root of 3 straight x squared space minus 2 square root of 2 straight x space minus 2 square root of 3 space equals space 0$

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353.

Find the roots of the quadratic equation $square root of 2 straight x squared space plus 7 straight x space plus 5 square root of 2 space equals space 0$

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354.

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

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355.

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

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Long Answer Type

356.

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.

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Short Answer Type

357.

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

358.
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.

Let the usual speed of the plane be x km/hr

Time taken to cover 1500 km with usual speed, t₁ = $\frac{1500}{x} h r$

Time taken to cover 1500 km with speed of (x +100) km/hr, t₂ $= \frac{1500}{x + 100} h r s$

Difference between t₁ - t₂ =30 min = 1/2

$∴ \frac{1500}{x} = \frac{1500}{x + 100} + \frac{1}{2} \frac{1500}{x} - \frac{1500}{x + 100} = \frac{1}{2} 1500 (\frac{x + 100 - x}{x (x + 100)}) = \frac{1}{2} 150000 x 2 = x (x + 100) x^{2} + 100 x - 300000 = 0 s o l v i n g e q u a t i o n, x = - 600 o r x = 500$

But speed can't be negative
Hence, usual speed 500 km/hr.

Long Answer Type

359.

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.

360.

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 ?