x k x - 56 = - 16 In the given equation, k is an integer constant. If the equation

Question

x k x - 56 = - 16 In the given equation, k is an integer constant. If the equation has no real solution, what is the least possible value of k ?

Accepted Answer

The correct answer is 50 . An equation of the form a x 2 + b x + c = 0 , where a , b , and c are constants, has no real solutions if and only if its discriminant, b 2 - 4 a c , is negative. Applying the distributive property to the left-hand side of the equation x k x - 56 = - 16 yields k x 2 - 56 x = - 16 . Adding 16 to each side of this equation yields k x 2 - 56 x + 16 = 0 . Substituting k for a , - 56 for b , and 16 for c in b 2 - 4 a c yields a discriminant of - 56 2 - 4 k 16 , or 3,136 - 64 k . If the given equation has no real solution, it follows that the value of 3,136 - 64 k must be negative. Therefore, 3,136 - 64 k < 0 . Adding 64 k to both sides of this inequality yields 3,136 < 64 k . Dividing both sides of this inequality by 64 yields 49 < k , or k > 49 . Since it's given that k is an integer, the least possible value of k is 50 .

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