Find the values of $k$ for which the following equations have real roots.
$k x (x - 2 \sqrt{5}) + 10 = 0$

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Solution

$k x (x - 2 \sqrt{5}) + 10 = 0$
$k x^{2} - 2 \sqrt{5} k x + 10 = 0$ and $k \neq 0$
This is in the form of $a x^{2} + b x + c = 0$
Here $a = k, b = - 2 \sqrt{5} k, c = 10$
For the equation to have real roots ,
discriminant $\geq 0 ⟹ (- 2 \sqrt{5} k)^{2} - 4 (k) (10) \geq 0$
$⟹ 20 k^{2} - 40 k \geq 0$
$⟹ k (k - 2) \geq 0 ⟹ k \in (- \infty, 0) \cup [2, \infty)$

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Find the values of k for which the following equations have real roots.kx(x−2√5)+10=0

kx(x−2√5)+10=0kx2−2√5kx+10=0 and k≠0This is in the form of ax2+bx+c=0Here a=k,b=−2√5k,c=10For the equation to have real roots ,discriminant ≥0⟹(−2√5k)2−4(k)(10)≥0⟹20k2−40k≥0⟹k(k−2)≥0⟹k∈(−∞,0)∪[2,∞)

Find the values of $k$ for which the following equations have real roots.
$k x (x - 2 \sqrt{5}) + 10 = 0$