If α+β are the roots of x²-(K+6)*x+2(2k-1),find the value of k if α+β =1/2 αβ

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If α+β are the roots of x²-(K+6)*x+2(2k-1),find the value of k if α+β =1/2 αβIf α+β are the roots of x²-(K+6)*x+2(2k-1),find the value of k if α+β =1/2 αβ

Answer 1

The roots are α and β, so (x-α)(x-β)=x²-(α+β)x+αβ=x²-(k+6)x+2(2k-1).

Equating coefficients: k+6=α+β and 2(2k-1)=4k-2=αβ.

Since α+β=½αβ, k+6=½αβ, so αβ=2(k+6); 4k-2=αβ, so 4k-2=2k+12, 2k=14, k=7.

[The quadratic becomes: x²-13x+26.]