Math, asked by Anonymous, 8 months ago

40.if a and b are roots of equation ax^2 +bx +c = 0 ,show that 2a ,2b are roots of the equation ax^2 +2bx +4c​

Answers

Answered by Joker444
5

Answer:

Given:

  • If \alpha and \beta are roots of equation ax² + bx + c = 0

To prove:

  • 2\alpha and 2\beta are the roots of the equation ax² + 2bx + 4c = 0

Solution:

The given equation is ax² + bx + c = 0

\boxed{\it{Sum \ of \ roots=-b/a}} \\ \\ \sf{\therefore{\alpha+\beta=-b/a}} \\ \\ \textsf{Multiply both sides by 2, we get} \\ \\ \sf{\therefore{2\alpha+2\beta=-2/a...(1)}} \\ \\ \boxed{\it{Product \ of \ roots=c/a}} \\ \\ \sf{\therefore{\alpha\beta=c/a}} \\ \\ \textsf{Multiply both sides by 4, we get} \\ \\ \sf{\therefore{4\alpha\beta=4c/a...(2)}} \\ \\ \sf{If \ roots \ were \ 2\alpha \ and \ 2\beta \ equation} \\ \\ \sf{can \ be \ written \ as} \\ \\ \sf{x^{2}-(2\alpha+2\beta)x+(2\alpha\times2\beta)=0} \\ \\ \sf{\therefore{x^{2}-(2\alpha+2\beta)x+(4\alpha\beta)=0}} \\ \\ \sf{From \ (1) \ and \ (2), \ we \ get} \\ \\ \sf{x^{2}+\dfrac{2bx}{a}+\dfrac{4c}{a}=0} \\ \\ \sf{Multiply \ throughout \ by \ a \ we \ get} \\ \\ \sf{ax^{2}+2b+4c=0} \\ \\ \sf{Hence, \ proved} \\ \\ \sf{Roots \ of \ ax^{2}+2b+4c=0 \ are \ 2\alpha \ and \ 2\beta}

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