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Step-by-Step Solution
Step 1: Identify the given equation and its roots
We have the quadratic equation
x^2 + 5\sqrt{2}\,x + 10 = 0.
Let its roots be \alpha and \beta , with \alpha > \beta .
We also define
P_n = \alpha^n - \beta^n.
Step 2: Recognize the expression to be simplified
We want to find the value of
\displaystyle \frac{P_{17} \, P_{20} + 5\sqrt{2}\,P_{17} \, P_{19}}{P_{18} \, P_{19} + 5\sqrt{2}\,P_{18}^2}.
Step 3: Factor out common terms
Factor P_{17} from the numerator and P_{18} from the denominator:
\displaystyle
\frac{P_{17} (P_{20} + 5\sqrt{2}\,P_{19})}{P_{18} (P_{19} + 5\sqrt{2}\,P_{18})}.
Step 4: Rewrite P_n in terms of powers of \alpha and \beta
Recall that
P_{n} = \alpha^n - \beta^n.
Hence,
P_{20} = \alpha^{20} - \beta^{20}, \quad P_{19} = \alpha^{19} - \beta^{19},
P_{18} = \alpha^{18} - \beta^{18}.
Thus the numerator becomes
P_{17} \Bigl((\alpha^{20} - \beta^{20}) + 5\sqrt{2}\,(\alpha^{19} - \beta^{19})\Bigr),
and the denominator becomes
P_{18} \Bigl((\alpha^{19} - \beta^{19}) + 5\sqrt{2}\,(\alpha^{18} - \beta^{18})\Bigr).
Step 5: Factor powers of \alpha and \beta
Notice that
\alpha^{20} = \alpha^{19}\,\alpha, \quad \beta^{20} = \beta^{19}\,\beta,
so we can factor these out:
\alpha^{20} - \beta^{20}
= \alpha^{19}(\alpha) - \beta^{19}(\beta),
5\sqrt{2}\,\alpha^{19} - 5\sqrt{2}\,\beta^{19}
= 5\sqrt{2}\,\bigl(\alpha^{19} - \beta^{19}\bigr).
Hence we rewrite the numerator as
P_{17}\bigl(\alpha^{19}(\alpha + 5\sqrt{2}) - \beta^{19}(\beta + 5\sqrt{2})\bigr).
A similar factorization in the denominator yields
P_{18}\bigl(\alpha^{18}(\alpha + 5\sqrt{2}) - \beta^{18}(\beta + 5\sqrt{2})\bigr).
Step 6: Use the quadratic relation to simplify \alpha + 5\sqrt{2} and \beta + 5\sqrt{2}
From the quadratic equation
\alpha^2 + 5\sqrt{2}\,\alpha + 10 = 0,
we can solve for \alpha + 5\sqrt{2} :
\alpha^2 = -5\sqrt{2}\,\alpha - 10 \quad\Longrightarrow\quad
\alpha + 5\sqrt{2} = -\frac{10}{\alpha}.
Similarly,
\beta^2 + 5\sqrt{2}\,\beta + 10 = 0 \quad\Longrightarrow\quad
\beta + 5\sqrt{2} = -\frac{10}{\beta}.
Step 7: Substitute and simplify
Substitute \alpha + 5\sqrt{2} = -\frac{10}{\alpha} and \beta + 5\sqrt{2} = -\frac{10}{\beta} into the factorized form.
This produces factors of -\frac{10}{\alpha} and -\frac{10}{\beta} , which will cancel with the powers of \alpha and \beta in the respective terms.
After cancellation, the entire expression simplifies down to
1.
Step 8: Conclude the final value
Therefore, the value of
\displaystyle \frac{P_{17} P_{20} + 5\sqrt{2}\,P_{17}\,P_{19}}{P_{18}\,P_{19} + 5\sqrt{2}\,P_{18}^2}
is
1.
