If the set of all a ∈ R , for which the equation 2 x 2 + ( a − 5 ) x + 15 = 3 a has no real root, is the interval ( α , β ) , and X = { x ∈ Z : α

Question

If the set of all a ∈ R , for which the equation 2 x 2 + ( a − 5 ) x + 15 = 3 a has no real root, is the interval ( α , β ) , and X = { x ∈ Z : α < x < β } , then ∑ x ∈ X ​ x 2 is equal to :

TestHub · Accepted Answer

​ ( a − 5 ) 2 − 8 ( 15 − 3 a ) < 0 a 2 + 14 a + 25 − 120 < 0 a 2 + 14 a − 95 < 0 ( a + 19 ) ( a − 5 ) < 0 a ∈ ( − 19 , 5 ) ∴ − 19 < x < 5 ∴ x ∈ X ∑ ​ x 2 = ( 1 2 + 2 2 + … . + 4 2 ) + ( 1 2 + 2 2 + … + 1 8 2 ) = 6 4 × 5 × 9 ​ + 6 18 × 19 × 37 ​ = 30 + 2109 = 2139 ​

Mathematics - Quadratic Equation Question with Solution | TestHub

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