Question

. If α and β are the roots of

2x^2+7x-9=0 then find the equation whose roots are
α/β ,β/α

Answer 1

α/β= -2/9 β/α=-4.5

Explanation:

So we have quadratic equation 2x^2+7x-9=0

Lets fin the roots using the equation's discriminant:

D=b^2-4*a*c

a=2 (coef at x^2) b=7(coef at x) c=-9

D= 49+4*2*9=121

sqrt(D)=11

So x1= (-b+sqrt(D))/(2*a)

x1=(-7+11)/4=1 so α=1

x2=(-7-11)/4=-4.5 so β=-4.5

=>α/β= -2/9 => β/α=-4.5

Answer 2

`18x^2+85x+18 = 0`

Explanation:

Given Equation is

=>
`2x^2+7x-9=0`

Comparing it with
`ax^2+bx+c = 0`, we get

=> a = 2, b = 7 and c = -9

So,

Sum of roots = α+β =
`-(b)/(a)`

α+β = -7/2

Product of roots = αβ = c/a

αβ = -9/2

Now, Finding the equation whose roots are:

α/β ,β/α

Sum of Roots =
`(\alpha )/(\beta ) + (\beta )/(\alpha )`

Sum of Roots =
`(\alpha^2+\beta^2 )/(\alpha \beta )`

Sum of Roots =
`((\alpha+\beta )^2-2\alpha\beta )/(\alpha\beta )`

Sum of roots =
`((-7)/(2) )^2-2((-9)/(2) ) / (-9)/(2)`

Sum of roots =
`(49)/(4) + 9 /(-9)/(2)`

Sum of Roots =
`(49+36)/(4) / (-9)/(2)`

Sum of roots =
`(85)/(4) * (2)/(-9)`

Sum of roots = S =
`-(85)/(18)`

Product of Roots =
`(\alpha )/(\beta ) (\beta )/(\alpha )`

Product of Roots = P = 1

The Quadratic Equation is:

=>
`x^2-Sx+P = 0`

=>
`x^2 - (-(85)/(18) )x+1 = 0`

=>
`x^2 + (85)/(18)x + 1 = 0`

=>
`18x^2+85x+18 = 0`

This is the required quadratic equation.