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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:

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

1147 Views

353.

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

3563 Views

354.

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

4084 Views

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.

Let the speed of the stream be x km/h.
It is given that the speed of a boat in still water is 15 km/h.
Now,
Speed of the boat upstream = Speed of the boat in still water − Speed of the stream = (15 − x) km/h
Speed of the boat downstream = Speed of the boat in still water + Speed of the stream = (15 + x) km/h
We know that
Time=Distance/Speed
Time is taken for upstream journey + Time taken for the downstream journey = 4 h 30 min

$therefore space fraction numerator 30 over denominator 15 minus straight x end fraction space plus fraction numerator 30 over denominator 15 plus straight x end fraction space equals space 4 1 half rightwards double arrow space 30 open square brackets fraction numerator open parentheses 15 plus straight x close parentheses plus open parentheses 15 minus straight x close parentheses over denominator open parentheses 15 plus straight x close parentheses open parentheses 15 minus straight x close parentheses end fraction close square brackets space equals space 9 over 2 rightwards double arrow space 30 space open parentheses fraction numerator 30 over denominator 225 minus straight x squared end fraction close parentheses space equals space 9 over 2 rightwards double arrow space 225 minus straight x squared space equals space fraction numerator 30 space straight x 30 space straight x 2 over denominator 9 end fraction rightwards double arrow space 225 minus straight x squared space equals space 200 rightwards double arrow space straight x squared space equals space 225 minus 200 space equals space 25 rightwards double arrow space straight x space equals space plus-or-minus 5$

Since speed can not be negative, therefore, x = 5.
Thus, the speed of the stream is 5 km/h.

7677 Views

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.

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 ?