Why is this approach wrong?

Question

If $\alpha$ and $\beta$ are the roots of $x^{2} - 4 x - 3$, then find $$\frac{1}{(\alpha-4)^{2}}+\frac{1}{(\beta-4)^{2}}$$

---

Solution

This is the approach that gives wrong answer.

\begin{array}{l} x^{2}-4 x-3=0 \\ (x-4)^{2}=19 \\ So, {(\alpha-4)^{2}} {\&(\beta-4)^{2}}=19 \\ \text { Hence, } \frac{1}{(\alpha-4)^{2}}+\frac{1}{(\beta-4)^{2}}=\frac{2}{19} \end{array}

This is how it is solved :

\begin{array}{l} x^{2}-4 x-3=0 \\ \alpha^{2}-4 \alpha-3=0 \\ \alpha(\alpha-4)=3 \\ \alpha-4=\frac{3}{\alpha} \\ \& \beta-4=\frac{3}{\beta} \end{array}

So , $$ \frac{1}{(\alpha-4)^{2}}+\frac{1}{(\beta-4)^{2}} $$ \begin{array}{l} =\frac{\alpha^{2}}{9}+\frac{\beta^{2}}{9} \\ =\frac{1}{9}\left(\alpha^{2}+\beta^{2}\right) \\ =\frac{22}{9} \end{array}

Please tell me why the first approach is wrong .

Answer 1

You did not factorise $x^2-4x-3$ correctly. Actually, $x^2-4x-3=(x-2)^2-4-3=(x-2)^2-7$. We find that $(x-2)^2=7$, and so $x-2=\pm\sqrt{7}$. Therefore, $x-4=-2\pm\sqrt{7}$, and $(x-4)^2=11\pm4\sqrt{7}$. Hence, $$ \frac{1}{(\alpha-4)^2}+\frac{1}{(\beta-4)^2}=\frac{1}{11+4\sqrt{7}}+\frac{1}{11-4\sqrt{7}}=\frac{11-4\sqrt{7}+11+4\sqrt{7}}{121-112}=\frac{22}{9} \, . $$ An alternative method (which would be slower in this case), would be to make the substitution $w=x-4$. The resulting quadratic in $w$ will have roots $\alpha-4$, and $\beta-4$.

Answer 2

$x^2-4x-3=(x-2)^2-7$

This is the mistake in your analysis.

Answer 3

Another approach would be to go ahead and perform the ratio addition (but don't fully multiply out the binomial-squares in the denominator):

$$ \frac{1}{(\alpha-4)^{2}}+\frac{1}{(\beta-4)^{2}} \ \ = \ \ \frac{(\beta-4)^{2} \ + \ (\alpha-4)^{2}}{(\alpha-4)^{2}· (\beta-4)^{2}} \ \ = \ \ \frac{\alpha^2 \ + \ \beta^2 \ - \ 8·(\alpha + \beta) \ + \ 32}{[ \ \alpha · \beta \ - \ 4·(\alpha + \beta) \ + 16 \ ]^2} $$ $ [ \ \alpha^2 + \beta^2 \ = \ (\alpha + \beta)^2 \ - \ 2·\alpha·\beta \ ] $

$$ = \ \ \frac{(\alpha \ + \ \beta)^2 \ - \ 2·\alpha·\beta \ - \ 8·(\alpha + \beta) \ + \ 32}{[ \ \alpha · \beta \ - \ 4·(\alpha + \beta) \ + 16 \ ]^2} $$ [we then apply Viete's relations: $ \ \alpha + \beta \ = \ -(-4) \ = \ 4 \ \ , \ \ \alpha·\beta \ = \ -3 \ ] $

$$ = \ \ \frac{4^2 \ - \ 2·(-3) \ - \ 8·(4) \ + \ 32}{[ \ (-3) \ - \ 4·4 \ + 16 \ ]^2} \ \ = \ \ \frac{16 \ + \ 6 \ - \ 32 \ + \ 32}{[ \ (-3) \ - \ 16 \ + 16 \ ]^2} \ \ = \ \ \frac{16 \ + \ 6 }{(-3)^2} \ \ = \ \ \frac{22}{9} \ \ . $$

This method also allows us to proceed without determining the zeroes of the polynomial themselves.