k x 2 − 1 7 x + 2 k = 0
Find the maximum integral value of k for which the above quadratic equation has two distinct real roots.
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Inicialmente, devemos julgar a condição, necessária, para que a equação em questão tenha duas raízes distintas.
Ora, sabemos que isso ocorre quando o valor do discriminante é maior que zero, isto e, Δ > 0 .
Isto posto, temos que:
Δ > 0 2 8 9 − 8 k 2 > 0 ( 6 , 0 1 + k ) ( 6 , 0 1 − k ) > 0 S = { k ∈ R ∣ − 6 , 0 1 < k < 6 , 0 1 }
Daí, o maior valor é 6.
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A quadratic equation has two distinct real roots if and only if its discriminant is greater than 0 .
Discriminant of k x 2 − 1 7 x + 2 k = ( − 1 7 ) 2 − 4 ( k ) ( 2 k ) = 2 8 9 − 8 k 2
So , 2 8 9 − 8 k 2 > 0 ⇒ k 2 < 8 2 8 9 ⇒ k 2 < 3 6 . 1 2 5
Since 6 2 = 3 6 , maximum required k = 6 .