So, the distance ‘x’ of the pole from gate B satisfies the equation

x 2 + 7x – 60 = 0

So, it would be possible to place the pole if this equation has real roots. To see if this is so or not, let us consider its discriminant. The discriminant is

b 2 – 4ac = 72 – 4 × 1 × (– 60) = 289 > 0.

So, the given quadratic equation has two real roots, and it is possible to erect the pole on the boundary of the park.

Solving the quadratic equation x 2 + 7x – 60 = 0, by the quadratic formula, we get

7 289 7 17 2 2 -± -± x = =

Therefore, x = 5 or – 12.

Since x is the distance between the pole and the gate B, it must be positive.

Therefore, x = – 12 will have to be ignored. So, x = 5.

Thus, the pole has to be erected on the boundary of the park at a distance of 5m from the gate B and 12m from the gate A

Example: 16

Find the discriminant of the equation 2 1 3 2 3 x x - + = 0 and hence find the nature of its roots. Find them, if they are real.

solution:

Here a = 3, b = – 2 and c = 1 3

Therefore, discriminant 2 2 1 4 ( 2) 4 3 4 4 0.

Hence, the given quadratic equation has two equal real roots.

The roots are , , 2 2 - - b b a a i.e., 2 2 , , 6 6 i.e., 1 1 , . 3 3



pgno.126


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