The value of a for which the sum of the squares of the roots of the equation x² - (a - 2)x - a - 1 =

Q: Wait, why can't we just solve for the roots using the quadratic formula and then square them?

That is a valid instinct, but it is a trap! If you use the quadratic formula, you will get roots involving square roots of expressions containing $a$. Squaring those expressions will lead to a very complicated algebraic mess that is prone to calculation errors. By using Vieta's formulas, we bypass the need to know the individual roots entirely, which is the elegant, professional way to solve this.

Q: Why does the identity $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$ work?

Think of it as expanding the square of a binomial. If you expand $(\alpha + \beta)^2$, you get $\alpha^2 + 2\alpha\beta + \beta^2$. To isolate $\alpha^2 + \beta^2$, you simply subtract the $2\alpha\beta$ term from both sides. It is a fundamental algebraic manipulation that turns a problem about individual roots into a problem about the sum and product of roots, which are directly given by the coefficients.

Q: How do I know if the quadratic function $S(a)$ has a minimum?

Look at the coefficient of the $a^2$ term in your final expression for $S(a)$. If the coefficient is positive, the parabola opens upwards, meaning it has a global minimum at its vertex. If it were negative, it would open downwards and have a maximum instead. Since our $S(a) = a^2 - 2a + 6$ has a positive coefficient for $a^2$, we are guaranteed a minimum.

Question

The value of $a$ for which the sum of the squares of the roots of the equation $x^2 - (a-2)x - a - 1 = 0$ assume the least value is

Options:

(A) $1$

(B) $0$

(C) $3$

(D) $2$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a quadratic equation: $x^2 - (a-2)x - (a+1) = 0$. At first glance, it looks like a standard problem, but the presence of the parameter $a$ makes it dynamic.

Many students immediately reach for the quadratic formula, hoping to find $x$. But stop! In the world of JEE Advanced, the most elegant path is rarely the most brute-force one.

We are looking for the sum of the squares of the roots, $S = \alpha^2 + \beta^2$. Instead of finding $\alpha$ and $\beta$ individually, we use the bridge built by Vieta.

Vieta's formulas tell us that for any quadratic $Ax^2 + Bx + C = 0$, the sum of the roots is $\alpha + \beta = -\frac{B}{A}$ and the product is $\alpha\beta = \frac{C}{A}$.

For our equation, $A=1$, $B=-(a-2)$, and $C=-(a+1)$. Thus, we have:

$$\alpha + \beta = a - 2$$

$$\alpha\beta = -(a + 1)$$

This is the secret key that unlocks the problem.

### The Algebraic Transformation

Now, we need to express $S = \alpha^2 + \beta^2$ in terms of $a$. We invoke the algebraic identity:

$$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$$

This is where the magic happens. By substituting our Vieta expressions, we get:

$$S = (a - 2)^2 - 2[-(a + 1)]$$

Let's expand this carefully. The term $(a - 2)^2$ becomes $a^2 - 4a + 4$. The term $-2[-(a + 1)]$ simplifies to $+2(a + 1)$, which is $2a + 2$.

Combining these, we get $S(a) = a^2 - 4a + 4 + 2a + 2$, which simplifies beautifully to:

$$S(a) = a^2 - 2a + 6$$

We have successfully transformed a problem about roots into a simple quadratic function of $a$.

### The Parabolic Insight

Now, we have $S(a) = a^2 - 2a + 6$. To find the minimum value, we visualize this as a parabola opening upwards. The minimum occurs at the vertex.

We can find this by completing the square:

$$S(a) = (a^2 - 2a + 1) - 1 + 6$$

$$S(a) = (a - 1)^2 + 5$$

Since $(a - 1)^2$ is always non-negative, the smallest value it can take is $0$, which happens when $a = 1$.

Thus, the **minimum value of $S$ is $5$**, and it occurs when $a = 1$.

This is the power of algebraic insight—by manipulating the structure of the equation rather than solving it, we find the answer with clarity and confidence. Keep this mindset, and you will conquer any quadratic problem the JEE throws at you.

Animated Solution for Mathematics - Quadratic Equations: The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assume the least value is

Select Answer:

Visualized Solution

Analyze the Equation

Apply Vieta's Formulas

Extract Coefficients

Define the Target Function

Substitute the Values

Expand the Terms

Simplify the Expression

Strategy to Minimize

Complete the Square

Analyze the Minimum

Final Conclusion

The Sigma Insight: Relation Between Roots and Coefficients

Analyzing the Setup

The Algebraic Transformation

The Parabolic Insight

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