Question

if a,B are the roots of x²–4x+3 the find the a²+ B² and a³+B³​

Answer 1

Answer:

$\huge\boxed{\fcolorbox{blue}{orange}{HELLO\:MATE}}$

Step-by-step explanation:

GIVEN:

$\alpha$&$\beta$are the roots of x²–4x+3.

TO FIND:

$\alpha^{2}$+$\beta^{2}$ and a³+B³

ANSWER:

The given equation is a Quadratic equation.

So, we can find it's roots by either Factorising or by standard formula.

On Factorising:

$x^{2}-4x+3$

=$x^{2}-x-3x+3$

=$x(x-1) -3(x-1)$

=$(x-3) (x-1)$

On equating with 0.

$(x-3) (x-1) =0$

Therefore x = 3,1.

$\large\red{\boxed{ x =2, 1}}$

$\alpha=3$

$\beta=1$

So, $\alpha^{2}$ = 3×3=9

$\beta^{2}$ =1×1=1

Hence, $\large\purple{\boxed{\alpha^{2}+ \beta^{2} =9+1=10}}$

And,

$\alpha^{3}$ =3×3×3=27

$\alpha^{3}$ =3×3×3=27

$\beta^{3}$=1×1×1=1

Hence

$\large\green{\boxed{\alpha^{3}+\beta^{3}=27+1=28.}}$

We can also find roots by using standard formula

that is

$\large\blue{\boxed{\dfrac{-b+-\sqrt{b^{2}-4ac}}{2a}}}$

Answer 2

Answer:

$\huge{\boxed{\fcolorbox{red}{pink}{Answer}}}$

given =

$\alpha \: and \: \beta \: are \: the \: roots \: of \: {x}^{2} - 4x + 3$

to find =

${ \alpha }^{2} + { \beta }^{2} \: and \: { \alpha }^{3} + { \beta }^{3}$

then given equation is quadratic equation . so we can find the roots by either factorising by the standard formulae...

on factorising :-

${x}^{2} - 4x + 3 = 0 \\ \\ {x}^{2} - x - 3x + 3 = 0 \\ \\ x(x - 1) - 3(x - 1) \\ \\ (x - 1)(x - 3)$

on equating with 0

$(x - 3)(x - 1) = 0$

therefore x = 3 , 1

$\boxed{\red{\mathfrak{x \: =\: 2 ,\: 1}}}$

$\alpha \: = 3 \\ \\ \beta \: = 1$

so

${ \alpha }^{2} = \: 9 \\ \\ { \beta }^{2} = 1$

hence

$\boxed{\purple{\mathfrak{{\alpha}^2 {+} {\beta}^2 {=} { 9 + 1 = 10}}}}$

and

${ \alpha }^{3} = 27 \\ \\ { \beta }^{3} = 1$

hence

$\boxed{\green{\mathfrak{\alpha}^3 {+} {\beta}^3 {=} { 27 + 1 = 28}}}$

we can also find roots by this formulae

${\boxed{\blue{\mathfrak{\frac{-b +- √{b}^2 - 4ac}{2a}}}}}$