Tag: vieta’s formula for quadratic equations

Let $\alpha ,\beta $ are roots of $2{ x }^{ 2 }-3x+5=0$ 
Then to get equation whose roots are $\displaystyle \dfrac { 1 }{ \alpha  } ,\dfrac { 1 }{ \beta  } $ 
Replace $\displaystyle x\rightarrow \frac { 1 }{ x } \ $
We get $5{ x }^{ 2 }-3x+2=0$.

$ The\quad roots\quad of\quad the\quad equation\quad { x }^{ 2 }+11x+13=0\quad is\quad diminished\quad by\quad 4\quad i.e.\ the\quad variable\quad becomes\quad (x+2).\ \therefore \quad
The\quad new\quad equation\quad is\quad \ (x+4)^{ 2 }+11(x+4)+13=0\ \Rightarrow { x }^{ 2 }+8x+16+11x+44+13=0\ \Rightarrow { x }^{ 2 }+19x+73=0\ Ans-\quad Option\quad C .$

For the equation, sum of roots $ = \alpha  + \beta = -\dfrac {3}{1} = -3 $
Product of roots $ = \alpha  \times \beta = \dfrac {3}{1} = 3 $

So, the eqn with roots $ = \alpha  + \beta$ and $ \alpha  \times \beta $ is $ (x - (-3))(x-3) = 0 $
$ => x^{2} -9 = 0 $

Given, $(x-2)(x^{2}+6x-11)=0$
$x^{3}+6x^{2}-11x-2x^{2}-12x-22=0$
$x^{3}+4x^{2}-23x-22=0$
Then $a=1  ,b=4  g=-22$
Sum of roots $(a+b+g) =\displaystyle \frac{-b}{a}=\frac{-4}{1}=-4$

The equation can be written as $x^{ 2 }+px+q=0$

$x^2 -3x = 3 = 0$ ........ (1)
Here, $a +\beta =3$ and $a\beta =3$. Therefore,
$2(a +\beta ) =6$
$2\times 2a\beta = 4a\beta =4 \times 3= 12$
The equation whose roots are double of (1),
$x^2 -$(sum of the roots)x + product of the roots $=0$
will be $x^2 -6x + 12 =0$

The given equation is $x^2 -x + 1 = 0$ ....... (1)
Here, $a +\beta = 1$ and $a\beta = 1$. Therefore,
$a^2+\beta^2 = (a + \beta)^2 -2a\beta = 1-2= -1$
and $a^2\beta^2 = (a\beta)^2 = 1^2= 0$
Therefore, the equation whose roots are square of 1 is $x^2$ -(sum of the roots)x +product $=0$
or $x^2-(-1)x+ 1 =0$
or $x^2+x+ 1 =0$

$(x+p)(x+q)-k=0\ \Longrightarrow { x }^{ 2 }+(p+q)x+pq-k=0$

${ x }^{ 2 }+p=0$

let $p,q$ be roots of equation $ax^2+bx+c=0$