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Find the Values Of K For Which the Roots Are Real and Equal in Each of the Following Equation: X2 - 2(5 + 2k)X + 3(7 + 10k) = 0 - CBSE Class 10 - Mathematics

Question

Find the values of k for which the roots are real and equal in each of the following equation:

x2 - 2(5 + 2k)x + 3(7 + 10k) = 0

Solution

The given quadric equation is x2 - 2(5 + 2k)x + 3(7 + 10k) = 0, and roots are real and equal

Then find the value of k.

Here, a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)

As we know that D = b2 - 4ac

Putting the value of a = 1, b = -2(5 + 2k) and c = 3(7 + 10k)

= (-2(5 + 2k))2 - 4 x (1) x 3(7 + 10k)

= 4(25 + 20k + 4k2) - 12(7 + 10k)

= 100 + 80k + 16k2 - 84 - 120k

= 16 - 40k + 16k2

The given equation will have real and equal roots, if D = 0

Thus,

16 - 40k + 16k2 = 0

8(2k2 - 5k + 2) = 0

2k2 - 5k + 2 = 0

Now factorizing of the above equation

2k2 - 5k + 2 = 0

2k2 - 4k - k + 2 = 0

2k(k - 2) - 1(k - 2) = 0

(k - 2)(2k - 1) = 0

So, either

k - 2 = 0

k = 2

Or

2k - 1 = 0

2k = 1

k = 1/2

Therefore, the value of k = 2, 1/2.

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Solution Find the Values Of K For Which the Roots Are Real and Equal in Each of the Following Equation: X2 - 2(5 + 2k)X + 3(7 + 10k) = 0 Concept: Nature of Roots.
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