Trying to find the right answer to this..

The given equation is x 2 +k(2x+k−1)+2=0. ⇒x 2 +2kx+k(k−1)+2=0 So, a = 1, b = 2k, c = k(k − 1) + 2 We know D=b 2 −4ac ⇒D=(2k) 2 −4×1×[k(k−1)+2] ⇒D=4k 2 −4[k2−k+2] ⇒D=4k 2 −4k2+4k−8 ⇒D=4k−8=4(k−2) For equal roots, D = 0 Thus, 4(k − 2) = 0 So, k = 2.

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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.

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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.

The given equation is x²+k(2x+k−1)+2=0.
⇒x²+2kx+k(k−1)+2=0
So, a = 1, b = 2k, c = k(k − 1) + 2
We know D=b²−4ac
⇒D=(2k)²−4×1×[k(k−1)+2]
⇒D=4k²−4[k2−k+2]
⇒D=4k²−4k2+4k−8
⇒D=4k−8=4(k−2)
For equal roots, D = 0
Thus, 4(k − 2) = 0
So, k = 2.

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355.

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Long Answer Type

356.

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Short Answer Type

357.

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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.

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 ?