under what conditions is 2x² + kx + 2 always positive ? solve the question
Answers
Step-by-step explanation:
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Therefore the given quadratic equation is positive for -5 < k < 4.
Given:
The quadratic equation: 2x² + kx + 2 = 0
To Find:
The conditions under which given function 2x² + kx + 2 = 0 is always positive.
Solution:
The given question can be answered very easily as shown below.
Given quadratic equation, 2x² + kx + 2 = 0
If a quadratic equation ax² + bx + c = 0 is given,
Then the condition for the function to be always positive is,
⇒ Determinant = b² - 4ac < 0 and f(a) > 0
Now comparing with the given equation,
a = 2, b = k, and c = 2
⇒ Determinant = b² - 4ac = k² - ( 4 × 2 × 2 ) = k² - 16
Conditions for the function to be positive,
Condition-1:
⇒ b² - 4ac = k² - 16 < 0
⇒ k < 4
Condition-2:
⇒ f(a) > 0 where a = 2
⇒ 2 (2)² + k (2) + 2 > 0
⇒ 10 + 2k > 0
⇒ k > -5
So from both the results, to get the positive value of the equation,
-5 < k < 4.
Therefore the given quadratic equation is positive for -5 < k < 4.
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