Question

If (2x-3) is a factor of 2x3 - 3x2 +15x -15k, find the value of 3k – √5k

Answer 1

ATQ, (2x - 3) is a factor of 2x³ - 3x² + 15x - 15k

given p(x) = 2x³ - 3x² + 15x - 15k

g(x) = 2x - 3 = 0

==> 2x = 3

==> x = 3/2

on putting values :-

p(3/2) = 2(3/2)³ - 3(3/2)² + 15(3/2) - 15k = 0

= 2(27/8) - 3(9/4) + 45/2 - 15k = 0

= 27/4 - 27/4 + 45/2 - 15k = 0

= 45/2 - 15k = 0

= 45/2 - 30k/2 = 0

= 45 - 30k = 2 × 0

= 45 - 30k = 0

= -30k = -45

= k = -45/-30

=> k = 3/2

now, we have to find the value of 3k - √5k

by solving, we got k = 3

put k = 3/2

= 3(3/2) - √(5×3/2)

= 9/2 - 3√5/2

= (9 - 3√5)/2

hence, value of 3k - √5k is (9 - 3√5)/2

HOPE THIS HELPS..!!

Answer 2

Given Equation is f(x) = 2x^3 - 3x^2 + 15x - 15k.

By factor theorem, 2x - 3 is a factor of f(x), if f(3/2) = 0.

Plug x = 3/2, we get

= > 2(3/2)^3 - 3(3/2)^2 + 15(3/2) - 15k = 0

= > 45/2 - 15k = 0

= > 45/2 = 15k

= > 45/30 = k

= > k = 3/2.

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Now,

$= > 3k -\sqrt{5}k$

$= > 3 *\frac{3}{2} -\sqrt{5} *\frac{3}{2}$

$= > \frac{9 - 3\sqrt{5}}{2}$

Hope it helps!