Question

Write a quadratic polynomial, the sum and product of whose zeros are 5 and 6​

Answer 1

Answer:

x²-5x+6

Step-by-step explanation:

Given that α+β=5 and αβ=6

Req. Polynomial

= k[x²-(α+β)x-αβ]

=k[x²-5x+6]

=x²-5x+6

             

Answer 2

$\boxed {\underline {\mathbb {FINAL\:ANSWER:-}}}$

$\boxed {x^{2}-5x+6}$

$\boxed {\underline {\mathbb {GIVEN:-}}}$

the sum and product of whose zeros are 5 and 6

$\boxed {\underline {\mathbb {TO\:FIND:-}}}$

What’s the quadratic polynomial  

$\boxed {\underline {\mathbb {THINGS\:TO\:ASSUME:-}}}$

Let the variable of quadratic polynomial as x

And one of its zero as $\alpha$

And another zero as $\beta$

$\boxed {\underline {\mathbb {FORMULA\:USED:-}}}$

$\alpha + \beta = \frac{- coffecient \: of\: x }{ coffecient \: of\: x^{2} } \\\implies \alpha + \beta = \frac{-b}{a}$

$\alpha \beta = \frac{constant\:term }{ coefficient \: of\: x^{2} } \\\implies \alpha + \beta = \frac{c}{a}$

$\boxed {\underline {\mathbb {SOLUTION:-}}}$

General equation is $ax^{2}+bx+c$

Here,

$a= coefficient \: of\: x^{2}$

$b= coefficient \: of\: x$

$c=constant \:term$

as we have given that sum of zeroes $(\alpha + \beta )= 5$

and

product of zeroes $(\alpha \beta ) = 6$

we know that

$\alpha + \beta = \frac{-b}{a} = \frac{-(-5)}{1} \implies \frac {5}{1} \implies 5$

$\alpha \beta = \frac{c}{a} = \frac{6}{1} \implies 6$

Hence all conditions are proved  

We have taken b as -5 and c as 6 and a as 1 to satisfy the conditions  

Therefore $\boxed {a=1\:\:\:\:\:b=-5\:\:\:\:\:c=6}$

Now put all in the general form i.e. $ax^{2}+bx+c$

$Quadratic\:equation \implies \boxed {x^{2}-5x+6}$