Question

if α and β are the zeroes of the polynomial f(x)=2x²-5x+7,find the polynomial whose zeroes are 2α+3β and 3α+2β

Answer 1

2x^2-7x+2x+7=0
x(2x-7)+1(2x+7)=0
So x=7/2 or -7/2
therefore
$\alpha = \frac{7}{2}$
$\beta = - \frac{7}{2}$
then
$2 \alpha = 7$
$2 \beta = - 7$

Answer 2

D = 25 - 56 < 0
thus alpha and beta doesn't exist

for 2x² - 5x - 7 = 0
2x² -7x + 2x -7 = 0
2x(x+1) -7(x+1) = 0
x = 7/2 and -1

for roots a and b quadratic equation is
(x-a)(x-b) = 0

now
a = 2(7/2) + 3(-1) = 4
b = 21/2 -2 = 17/2

(x - 17/2)(x - 4) = 0