Question

alpha and beta are the roots of the equation X square - m x +n is equal to zero if Alpha is replaced by alpha + 2 and beta bye beta minus 2 then the equation becomes X square + bx + c is equal to zero and also if alpha and beta are replaced by Alpha minus 2 and beta minus 2 then the equation becomes X square + px + q=0

Answer 1

Step-by-step explanation:

alpha and beta are solutions of ax2+bx+c=0, so,

alpha+beta=−b/a

alpha.beta=c/a

This is the formula to calculate sum and product of roots of any quadratic equation.

If, −1/alpha and −1/beta are solutions of a quadratic equation, then, to find quadratic equation, which would be something like,

[x−(−1/alpha)].[x−(−1/beta)]=0

Implies,

[x+(1/alpha)].[x+(1/beta)]=0

Remove brackets,

x2+x.[(1/alpha)+(1/beta)]+1/alpha.beta=0

Take LCM for 1/alpha and 1/beta, which is alpha.beta, simplify, we get,

x2+x.[alpha+beta/alpha.beta]+1/alpha.beta=0 …..(1)

We know that,

Sum of roots of a Quadratic Equation of general form, i.e., ax2+bx+c=0, will always be −b/a

Product of roots of a Quadratic Equation of general form, i.e., ax2+bx+c=0, will always be c/a

So, from (1), we can write,

x2+x.[(−b/a)/(c/a)]+1/(c/a)=0

Since, the roots of ax2+bx+c=0 are alpha and beta.

Simplify, we get,

x2−bx/c+a/c=0

Take c as LCM, we get,

cx2−bx+a=0

I hope my answer was helpful.

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