If the set of all a ∈ R for which the equation 2x^2 + (a - 5)x + 15 = 3a has no real root, is the in

Q: Why is the discriminant strictly less than zero? Can't it be equal to zero?

That is a fantastic question. If the discriminant $D$ were equal to zero, the quadratic equation would have exactly one real root (a repeated root). Since the problem explicitly states that the equation has *no* real roots, we must exclude the case where the parabola touches the x-axis. Thus, $D < 0$ is the only condition that keeps the parabola floating entirely above or below the x-axis.

Q: Why do we use the wavy curve method for the inequality?

The wavy curve method, or the method of intervals, is your most reliable tool for solving polynomial inequalities. It allows you to visualize the sign changes of the expression across the critical points without having to test every single value. It is elegant, fast, and minimizes the chance of making a sign error.

Q: Is there a faster way to sum the squares than adding them one by one?

Absolutely! In JEE Advanced, time is your most precious resource. Instead of manual addition, we use the standard summation formula $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$. This formula is a powerful shortcut that turns a tedious calculation into a simple algebraic substitution.

Question

If the set of all $a \in R$ for which the equation $2x^{2}+(a-5)x+15=3a$ has no real root, is the interval $(\alpha,\beta)$, and $X=\{x \in Z:\alpha<x<\beta\}$ then $\sum_{x \in X}x^{2}$ is equal to :

Options:

(A) 2109

(B) 2129

(C) 2119

(D) 2139

Sigma Percentile · Accepted Answer

### Analyzing the Setup

When you look at a quadratic equation like $2x^2 + (a-5)x + 15 = 3a$, do not see it as a static collection of symbols. See it as a dynamic system where the variable $a$ is a parameter that shifts the entire parabola.

Our goal is to find the range of $a$ that forces this parabola to never touch the $x$-axis.

### The Foundation

Every great solution begins with clarity. The equation $2x^2 + (a-5)x + 15 = 3a$ must be brought into the standard form $Ax^2 + Bx + C = 0$.

By subtracting $3a$ from both sides, we obtain:

$$2x^2 + (a-5)x + (15-3a) = 0$$

Now, the structure is revealed. Our coefficient $A$ is $2$, $B$ is $(a-5)$, and the constant term $C$ is $(15-3a)$. This is the bedrock upon which we will build our entire argument.

### The Gatekeeper

Why does a quadratic have no real roots? Geometrically, it means the parabola is "floating" and never crosses the $x$-axis. Algebraically, this is governed by the discriminant $D = B^2 - 4AC$.

If $D < 0$, the parabola is entirely detached from the axis. We substitute our coefficients into this condition:

$$(a-5)^2 - 4(2)(15-3a) < 0$$

Expanding $(a-5)^2$ gives $a^2 - 10a + 25$. Distributing the $-8$ into $(15-3a)$ gives $-120 + 24a$. Combining these, we arrive at the inequality:

$$a^2 + 14a - 95 < 0$$

### The Wavy Curve

We are now looking for the values of $a$ that satisfy $a^2 + 14a - 95 < 0$. We factorize this quadratic by finding two numbers that multiply to $-95$ and add to $14$, which are $19$ and $-5$.

Thus, we have:

$$(a+19)(a-5) < 0$$

Using the wavy curve method, we identify the critical points at $a = -19$ and $a = 5$. Since the parabola $a^2 + 14a - 95$ opens upwards, it is negative between the roots. Therefore, $a$ must lie in the interval $(-19, 5)$.

### The Grand Finale

We are tasked with finding the sum of the squares of all integers $x$ such that $-19 < x < 5$. The set of integers $X$ contains values from $-18$ to $4$.

We need to calculate the sum:

$$\sum_{x=-18}^{4} x^2$$

The square of a negative number is identical to the square of its positive counterpart. We can split our sum into the sum of squares from $1$ to $18$, the square of $0$, and the sum of squares from $1$ to $4$.

Using the formula $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$:

For $n=18$:

$$\sum_{k=1}^{18} k^2 = \frac{18(19)(37)}{6} = 3 	imes 19 	imes 37 = 2109$$

For $n=4$:

$$\sum_{k=1}^{4} k^2 = \frac{4(5)(9)}{6} = 30$$

Adding these together, the final result is:

$$2109 + 30 = \mathbf{2139}$$

Animated Solution for Mathematics - Quadratic Equations: If the set of all $a \in R$ for which the equation $2 x^{2} + (a - 5) x + 15 = 3 a$ has no real root, is the interval $(α, β)$ , and $X = {x \in Z : α < x < β}$ then $\sum_{x \in X} x^{2}$ is equal to :

Select Answer:

Visualized Solution

Standard Form of the Equation

Condition for No Real Roots

Setting up the Discriminant

Expanding the Inequality

Factorizing the Expression

Solving the Inequality

Identifying $α$ and $β$

Defining the Set $X$

Setting up the Summation

Splitting the Sum

The Sum of Squares Formula

Calculating the First Sum ( $n = 18$ )

Calculating the Second Sum ( $n = 4$ )

Final Addition

The Sigma Insight: Nature of Roots

Analyzing the Setup

The Foundation

The Gatekeeper

The Wavy Curve

The Grand Finale

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