Animated Solution for Mathematics - Quadratic Equations: Let $\alpha$ and $\beta$ be the roots of $x^2+\sqrt{3}x-16=0$, and $\gamma$ and $\delta$ be the roots of $x^2+3x-1=0$. If $P_n={\alpha }^n+{\beta }^n$ and $Q_n={\gamma }^n+{\delta }^n$, then $\frac{P_{25}+\sqrt{3}{P}_{24}}{2P_{23}}+\frac{Q_{25}-Q_{23}}{Q_{24}}$ is equal to

Beyond Brute Force

Welcome, my aspiring engineers. Today, we are going to dismantle a problem that, at first glance, looks like a nightmare of arithmetic. You see an expression involving the 25th, 24th, and 23rd powers of roots, and your instinct might be to reach for the quadratic formula.

But I want you to pause. Take a deep breath. In the world of JEE Advanced, the most complex-looking problems often have the most elegant, simple solutions hidden just beneath the surface. We are not here to calculate; we are here to observe.

Let us look at our first equation: $x^2+\sqrt{3}x-16=0$. If you try to find the roots $\alpha$ and $\beta$ using the quadratic formula, you get:

Imagine raising this to the 25th power. It is a path to madness!

Whenever you see high powers like $n,n-1,n-2$ appearing in a ratio, your mind should immediately jump to Newton's Sums. This theorem is the bridge between the coefficients of a polynomial and the power sums of its roots.

It tells us that for any quadratic equation $a{x}^2+bx+c=0$, if $S_n={\alpha }^n+{\beta }^n$, then the following recurrence relation must hold:

This is not just a formula; it is a structural property of the roots. It says that the $n$-th power is entirely determined by the two preceding powers. It is the DNA of the polynomial.

Let us apply this to our first equation: $x^2+\sqrt{3}x-16=0$. Here, $a=1$, $b=\sqrt{3}$, and $c=-16$. Plugging these into our recurrence relation, we get:

Now, look at the target expression: $\frac{P_{25}+\sqrt{3}{P}_{24}}{2P_{23}}$. Notice the indices? They are 25, 24, and 23. This is a perfect match for our recurrence relation if we set $n=25$.

Let us substitute $n=25$ into our relation:

Do you see it? The numerator of our target fraction, $P_{25}+\sqrt{3}{P}_{24}$, is sitting right there! We can rearrange this to isolate it:

Now, divide both sides by $2P_{23}$. The $P_{23}$ terms cancel out beautifully, leaving us with $\frac{16}{2}=8$. Just like that, the first half of our monster expression has collapsed into the number 8.

We are not done yet. We have the second part of the expression: $\frac{Q_{25}-Q_{23}}{Q_{24}}$. Our second equation is $x^2+3x-1=0$.

Again, we apply Newton's Sums. Here, $a=1$, $b=3$, and $c=-1$. The recurrence relation becomes:

We need to relate $Q_{25},Q_{24},$ and $Q_{23}$. Let us set $n=25$ again:

Look at the numerator we need: $Q_{25}-Q_{23}$. Let us rearrange our equation to isolate these terms:

Now, divide both sides by $Q_{24}$. The $Q_{24}$ terms cancel out, and we are left with $-3$.

We have reduced a terrifying algebraic expression into two simple integers. The first fraction is $8$, and the second fraction is $-3$. The problem asks for their sum:

And there it is. The answer is 5.

My dear student, I want you to reflect on this journey. We didn't fight the problem with brute force; we outsmarted it with structure. Whenever you face a problem that seems to require impossible calculations, stop and ask yourself: "Is there a recurrence? Is there a symmetry? Is there a theorem I am forgetting?"