find the smallest integer value of k for which 2x² + 5x + 3k is always positive for all real value of x
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You need write x^2+ka+k = 0 has 2 distinct real roots. Only an equation has roots & an expression like x^2+kx+k does not have any roots. Now for amswer -
The discriminant b^2–4ac of the equation is k^2–8k. This must be > 0
So k(k-8) >0. It can happen if both the terms k, k-8 are < 0 OR when both are >0
Both are less than zero
k<0 and k-8<0 => k<0 and k<8 or k<0 (lower of the values)
Both are > 0
k>0 and k-8>0 => k>8
The result is when k <0 or k>8 the roots are real and distinct
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