Question 22

If $$f(x + 1) = x^2 - 3x + 2$$ and if the roots of the equation $$f(x) = 0$$ are $$\alpha$$ and $$\beta$$, then the value of $$\alpha^2 + \beta^2$$ is

Correct Answer: 13

Solution

Putting $$y = x + 1$$, so that $$x = y - 1$$, gives $$f(y) = (y-1)^2 - 3(y-1) + 2 = y^2 - 5y + 6$$. The roots of $$y^2 - 5y + 6 = 0$$ are 2 and 3, so $$\alpha^2 + \beta^2 = 4 + 9 = 13$$.

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