The sum of all integral values of k (k ≠ 0) for which the equation 2/(x-1) - 1/(x-2) = 2/k in x has

Q: Wait, why do we reject the negative root here? Or why can't we just use energy conservation?

This isn't a physics problem, so energy conservation doesn't apply! We are dealing with a rational equation. The 'no real roots' condition comes from the discriminant of the resulting quadratic equation being negative. We don't reject negative roots; we solve for the range of $k$ that makes the discriminant negative.

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

Great question! If the discriminant were equal to zero, the quadratic equation would have exactly one real root (a repeated root). The problem explicitly asks for the equation to have *no* real roots, so we must ensure $D < 0$ to avoid any real solutions.

Q: How do I know if I should include the endpoints of the range for $k$?

Since the inequality is strict ($D < 0$), the values of $k$ that make $D=0$ are excluded. We only consider integers strictly between the roots of the $k$-quadratic. Always check the boundary conditions carefully!

Question

The sum of all integral values of $k (k 
eq 0)$ for which the equation $\frac{2}{x-1} - \frac{1}{x-2} = \frac{2}{k}$ in $x$ has no real roots, is .

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing before a complex rational equation:

$$\frac{2}{x-1} - \frac{1}{x-2} = \frac{2}{k}$$

Our goal is to find the sum of all integral values of $k$ for which this equation has no real roots.

### The Algebraic Cleanup

First, we simplify the left-hand side by finding a common denominator, which is $(x-1)(x-2)$. We combine the fractions as follows:

$$\frac{2(x-2) - 1(x-1)}{(x-1)(x-2)}$$

Expanding the numerator, we get $2x - 4 - x + 1$, which simplifies to $x-3$. The denominator expands to $x^2 - 3x + 2$.

Now, our equation is:

$$\frac{x-3}{x^2-3x+2} = \frac{2}{k}$$

### The Quadratic Bridge

To move forward, we eliminate the denominators by cross-multiplying:

$$k(x-3) = 2(x^2 - 3x + 2)$$

Expanding both sides, we get $kx - 3k = 2x^2 - 6x + 4$. Bringing everything to one side, we form the standard quadratic equation:

$$2x^2 - (6+k)x + (4+3k) = 0$$

For this equation to have no real roots, the discriminant $D$ must be strictly less than zero.

### The Discriminant's Verdict

The discriminant is defined as $D = b^2 - 4ac$. Here, $a = 2$, $b = -(6+k)$, and $c = 4+3k$.

Substituting these into the formula, we get:

$$D = [-(6+k)]^2 - 4(2)(4+3k) < 0$$

Expanding this, we have $(36 + 12k + k^2) - 8(4 + 3k) < 0$, which simplifies to $36 + 12k + k^2 - 32 - 24k < 0$. Combining like terms, we arrive at the elegant inequality:

$$k^2 - 12k + 4 < 0$$

### The Final Tally

To solve $k^2 - 12k + 4 < 0$, we find the roots of $k^2 - 12k + 4 = 0$ using the quadratic formula:

$$k = \frac{12 \pm \sqrt{144 - 16}}{2} = 6 \pm 4\sqrt{2}$$

Since $\sqrt{2} \approx 1.414$, the roots are approximately $6 \pm 5.656$, giving us the range $0.344 < k < 11.656$.

The integral values of $k$ in this range are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11$. Summing these integers, we use the formula $\frac{n(n+1)}{2}$ for $n=11$:

$$\frac{11 	imes 12}{2} = 66$$

The sum of all such integral values of $k$ is **66**.

Animated Solution for Mathematics - Quadratic Equations: The sum of all integral values of $k (k \neq = 0)$ for which the equation $\frac{2}{x - 1} - \frac{1}{x - 2} = \frac{2}{k}$ in $x$ has no real roots, is .

Enter Numerical Value:

Visualized Solution

Analyze the Given Equation

Simplify the Rational Expression

Form the Simplified Equation

Raw Setup: Cross Multiplication

Rearrange to Standard Quadratic Form

Condition for No Real Roots

Raw Setup: Discriminant Substitution

Atomic Compute: Expanding the Inequality

Simplify the Inequality in $k$

Find the Roots of the $k$ -Quadratic

Calculate Numerical Range for $k$

Identify Integral Values of $k$

Calculate the Final Sum

The Sigma Insight: Nature of Roots

Analyzing the Setup

The Algebraic Cleanup

The Quadratic Bridge

The Discriminant's Verdict

The Final Tally

Similar Questions

The number of integral values of m for which the equation $(1 + m^{2}) x^{2} - 2 (1 + 3 m) x + (1 + 8 m) = 0$ has no real root is :

Consider the equation $x^{2} + 4 x - n = 0$ , where $n \in [20, 100]$ is a natural number. Then the number of all distinct values of $n$ , for which the given equation has integral roots, is equal to

The number of all possible positive integral values of $α$ for which the roots of the quadratic equation, $6 x^{2} - 11 x + α = 0$ are rational numbers is :

Let $p, q \in {1, 2, 3, 4}$ . The number of equations of the form $p x^{2} + q x + 1 = 0$ having real roots is

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 :

Let $S = {α : lo g_{2} (9^{2 α - 4} + 13) - lo g_{2} (\frac{5}{2} \cdot 3^{2 α - 4} + 1) = 2}$ . Then the maximum value of $β$ for which the equation $x^{2} - 2 (\sum_{α \in S} α)^{2} x + \sum_{α \in S} (α + 1)^{2} β = 0$ has real roots, is

The set of all values of $k > - 1$ , for which the equation $(3 x^{2} + 4 x + 3)^{2} - (k + 1) (3 x^{2} + 4 x + 3) (3 x^{2} + 4 x + 2) + k (3 x^{2} + 4 x + 2)^{2} = 0$ has real roots is :

The least positive value of 'a' for which the equation, $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is ____________.

The least positive value of $a$ for which the equation, $2 x^{2} + (a - 10) x + \frac{33}{2} = 2 a$ has real roots is

If $x^{2} + (a - b) x + (1 - a - b) = 0$ where $a, b \in R$ then find the values of $a$ for which equation has unequal real roots for all values of $b$ .