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Quadratic Formula

Find the valu...

Question

Find the value of $^{'} K^{'}$ if $Q . E$
$(2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0$ has equal roots

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Solution

We have the given equation
$(2 k + 1) x^{2} + 2 (k + 3) x + (k + 5) = 0$
using the formula of $b^{2} - 4 a c = 0$ we get
$[2 (k + 3) x]^{2} - 4 \times (2 k + 1) (k + 5) = 0$ [On putting the value of $a, b a n d c$ from the above given equation]
$4 (k + 3) x^{2} - (2 k + 1) (k + 5) = 0$

$k^{2} + 9 + 6 k - (2 k^{2} + 10 k + k + 5) = 0$
$k^{2} + 9 + 6 k - 2 k^{2} - 11 k - 5 = 0$
$- k^{2} - 5 k + 4 = 0$
$k^{2} + 5 k - 4 = 0$ [multiply both side by the sign $- v e$ ]
Now again use the formula of $b^{2} - 4 a c = 0$ we get

$k = \frac{- 5 \pm \sqrt{5^{2} + 4 \times 1 \times 4}}{2 \times 1}$
$= \frac{- 5 \pm \sqrt{25 + 16}}{2}$
$= \frac{- 5 \pm \sqrt{41}}{2}$
Which is the required answer

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