If {x^2} + px + q = 0 is the quadratic equati | vedclass

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(D) Given that $a$ and $b$ are the roots of ${x^2} - 3x + 1 = 0$.From the properties of quadratic equations,the sum of roots $a + b = 3$ and the produ

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None of these

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(D) Given that $a$ and $b$ are the roots of ${x^2} - 3x + 1 = 0$.From the properties of quadratic equations,the sum of roots $a + b = 3$ and the product of roots $ab = 1$.The roots of the equation ${x^2} + px + q = 0$ are $(a - 2)$ and $(b - 2)$.Sum of roots: $(a - 2) + (b - 2) = -p$$(a + b) - 4 = -p$$3 - 4 = -p \Rightarrow -1 = -p \Rightarrow p = 1$.Product of roots: $(a - 2)(b - 2) = q$$ab - 2(a + b) + 4 = q$$1 - 2(3) + 4 = q$$1 - 6 + 4 = q \Rightarrow q = -1$.Thus,$(p, q) = (1, -1)$.Since this pair is not listed in options $A, B,$ or $C$,the correct answer is $D$.

If ${x^2} + px + q = 0$ is the quadratic equation whose roots are $a - 2$ and $b - 2$,where $a$ and $b$ are the roots of ${x^2} - 3x + 1 = 0$,then

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