If the roots of the equation x 2 - 4x - log 10 N = 0 are real, then what is the minimum value of N ?
0.0001
Given:
Roots of x 2 - 4x - log 10 N = 0 are real
Formula used:
Roots of quadratic equation ax 2+ bx + c = 0 are real
if D ≥ 0 i.e b 2- 4ac ≥ 0
And lf log aN = x
Then N = a x
Calculation:
We have the quadratic equation x 2 - 4x - log 10 N = 0
On comparing this equation with ax 2 + bx + c = 0, we get
a = 1, b = -4 and c = - log 10 N
Now, According to the question
D = b 2 - 4ac ≥ 0
⇒ (- 4) 2- 4 × 1 × (- log 10 N) ≥ 0
⇒ 16 + 4log 10 N ≥ 0
⇒ 4log 10 N ≥ -16
⇒ log 10 N ≥ - 4
⇒ N ≥ 10 -4
⇒ N ≥ 0.0001
∴ The minimum value of N is 0.0001.
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