Question

If a, b are the roots of x^2-px +q=0 then a/b+b/a =

Answer 1

Answer:

Step-by-step explanation:

Given:

$\alpha$ and $\beta$ are the roots of x^2-px +q=0

To Prove:

$\alpha /\beta$ and $\beta /\alpha$ are roots of the equation

Answer:

if  a, b are the roots of x^2-px +q=0

then,

sum of the roots= $\alpha$+$\beta$= -b/a = p

           $\alpha+\beta$ = p     .................................(1)

product of the roots =$\alpha *\beta$ = c/a = q

           $\alpha * \beta$ = q      ..................................(2)

we know that,

Quadratic Equation= $x^2$ + ($\alpha + \beta$)*x + $\alpha \beta$

Quadratic Equation = $x^2$ + px + q

$\alpha /\beta , \beta /\alpha$ are roots of the a quadratic equation, then the quadratic equation is given by,

(x- $\alpha /\beta$ ) ( x - $\beta /\alpha$) = 0

⇒$x^2$- ( $\alpha /\beta + \beta /\alpha$ )x + 1 = 0

$\alpha \beta x^2$ - [$(\alpha +\beta) ^2$ - 2$\alpha \beta$]x+1=0      ..............................(3)

Substituting (1),(2)  in (3), we get

$qx^2$ $( -p^2-2q)x+q$

⇒$qx^2+(2q-p^2)x+q$=0