Question

If alpha and beta are the zeros of the quadratic polynomial f (x) = 5x + 7, find a polynomial whose zeroes are 2alpha +3 beta and 3 alpha+ 2beta ?

please tell me the answer of this question
take a time but PLEASE tell me the full solution with answer​

Answer 1

$\huge\underline\purple{Answer}$

When roots are given, equation in quadratic form

x² −(α+β)x+αβ=0

Here roots are (2α+3β) and (3α+2β)

∴x² −(2α+3β+3α+2β)x+(2α+3β)(3α+2β)=0

x²−5(α+β)x+6α² +4αβ+4αβ+6β² =0

f(x)=2x² −5x+7

α+β= 5/2, αβ=7/2

∴α ²+β² =(α+β)²−2αβ

= 25/4−7= −3/4

${x}^{2} - 5( \frac{5}{2} )x + 6( - \frac{3}{4} ) + 13 \times \frac{7}{2} \\ = 0 \\ \\ = 2 {x}^{2} - 50x - 18 + 182 = 0 \\ 2 {x}^{2} - 50x - + 162 = 0 \\ \\ \\ = {x}^{2} - 25x + 82 = 0$

$\huge\fbox\purple{@park Jimin }$

Hope it helps u ☺️