Let α$\alpha$ and β$\beta$ be the roots of x² + √(3)x - 16 = 0$x^{2} + \sqrt{3}x - 16 = 0$, and γ$\gamma$ and δ$\delta$ be the roots of x² + 3x - 1 = 0$x^{2} + 3x - 1 = 0$. If Pₙ = αⁿ + βⁿ$P_{n} = \alpha^{n} + \beta^{n}$ and Qₙ = γⁿ + δⁿ$Q_{n} = \gamma^{n} + \delta^{n}$, then P₂₅ + √(3)P₂₄2P₂₃ + Q₂₅ - Q₂₃Q₂₄$\frac{P_{25} + \sqrt{3}P_{24}}{2P_{23}} + \frac{Q_{25} - Q_{23}}{Q_{24}}$ is equal to ______

Related Formula

X² = |X|²$X^2 = |X|^2$ Quadratic factorization: t² - 5t + 6 = (t-2)(t-3)$t^2 - 5t + 6 = (t-2)(t-3)$

Core Logic

Rewrite the equation taking |x - 1| = t$|x - 1| = t$, where t ≥ 0$t \geq 0$. Since (x-1)² = |x-1|²$(x-1)^2 = |x-1|^2$, the equation becomes:

Step 1: Solve for modulus

Since both roots are positive, both provide valid solutions for the modulus. |x - 1| = 2 and |x - 1| = 3$|x - 1| = 2 \quad \text{and} \quad |x - 1| = 3$

Step 2: Unpack x values

From |x - 1| = 2$|x - 1| = 2$:

From |x - 1| = 3$|x - 1| = 3$:

The roots are 3, -1, 4, -2$3, -1, 4, -2$.

Step 3: Sum of Roots

Pattern Recognition

In symmetric modulus equations f(|x-a|) = 0$f(|x-a|) = 0$, every valid root t$t$ generates twin solutions (a+t)$(a+t)$ and (a-t)$(a-t)$. The sum of each pair is perfectly 2a$2a$. If there are n$n$ distinct valid positive roots for t$t$, the sum of all x$x$-roots is exactly n × 2a$n \times 2a$. Here, n=2, a=1 ⇒ 2 × 2(1) = 4$n=2, a=1 \Rightarrow 2 \times 2(1) = 4$.

Chapter Mix

Class 11 Maths: Quadratic Equations

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Core Logic

Rewrite the series in summation form:

Step 1: Simplify the Equation

Substitute the standard summation formulas:

Divide the entire equation by n$n$:

Let the roots be α$\alpha$ and β$\beta$.

Step 2: Apply the Condition for Roots

Since the roots are two consecutive even integers, their difference is 2$2$.

Since n$n$ is a positive integer, n = 3$n = 3$.

Pattern Recognition

When dealing with equations where roots have a specific difference k$k$, instantly use D = a²k²$D = a^2k^2$. Here k=2$k=2$, so D=4a²$D=4a^2$.

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Class 11 Maths: Quadratic Equations Class 11 Maths: Sequence and Series

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Core Logic

Given f(x) = x² + 2ax + (3a + 10)$f(x) = x^2 + 2ax + (3a + 10)$. Since the coefficient of x²$x^2$ is 1 > 0$1 > 0$, the parabola opens upward. For 1$1$ to lie between the roots α$\alpha$ and β$\beta$, the value of the function at x = 1$x = 1$ must be strictly less than 0$0$. f(1) < 0$f(1) < 0$

Step 1: Evaluate Inequality

5a + 11 < 0$5a + 11 < 0$

So, a in (-∞, -(11)/(5))$a \in \left(-\infty, -\frac{11}{5}\right)$.

Pattern Recognition

When a specified value k$k$ lies between roots, a · f(k) < 0$a \cdot f(k) < 0$. If a>0$a>0$, this simplifies to f(k) < 0$f(k) < 0$. No need to check discriminant Δ > 0$\Delta > 0$ manually because a f(k) < 0$a f(k) < 0$ guarantees real distinct roots.

Chapter Mix

Class 11 Maths: Quadratic Equations

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Core Logic

Break the domain into intervals based on the critical points of the absolute values, which are x = -4$x = -4$ and x = -2$x = -2$. Intervals to test: Case I: x < -4$x < -4$ Case II: -4 ≤ x < -2$-4 \leq x < -2$ Case III: x ≥ -2$x \geq -2$

Step 1: Analyzing Case I

For x < -4$x < -4$: Both (x+4)$(x+4)$ and (x+2)$(x+2)$ are negative. The equation becomes:

Roots are x = -7 ± √(49 + 16)2 = -7 ± √(65)2$x = \frac{-7 \pm \sqrt{49 + 16}}{2} = \frac{-7 \pm \sqrt{65}}{2}$.

Check if they fall in the interval x < -4$x < -4$: √(65) ≈ 8.06$\sqrt{65} \approx 8.06$ x₁ = (-7 + 8.06)/(2) ≈ 0.53$x_1 = \frac{-7 + 8.06}{2} \approx 0.53$ (Rejected, not <-4$<-4$) x₂ = (-7 - 8.06)/(2) ≈ -7.53$x_2 = \frac{-7 - 8.06}{2} \approx -7.53$ (Accepted) So, 1 valid solution here.

Step 2: Analyzing Case II

For -4 ≤ x < -2$-4 \leq x < -2$: (x+4) ≥ 0$(x+4) \geq 0$ and (x+2) < 0$(x+2) < 0$. The equation becomes:

x² + x + 4 = 0$x^2 + x + 4 = 0$

Discriminant D = 1² - 4(1)(4) = 1 - 16 = -15 < 0$D = 1^2 - 4(1)(4) = 1 - 16 = -15 < 0$. No real roots in this interval.

Step 3: Analyzing Case III

For x ≥ -2$x \geq -2$: Both (x+4)$(x+4)$ and (x+2)$(x+2)$ are positive. The equation becomes:

Discriminant D = 49 - 64 = -15 < 0$D = 49 - 64 = -15 < 0$. No real roots in this interval.

Step 4: Final Count

Combining the results from all cases, there is exactly 1 distinct real solution.

Pattern Recognition

For sum-of-absolute-value equations, systematically partition the number line using the roots of the arguments. Discard roots generated by the quadratics that fall outside their respective assumed interval.

Chapter Mix

Class 11 Maths: Quadratic Equations

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Difference of roots formula:

For 12x² - 20x + 3λ = 0$12x^2 - 20x + 3\lambda = 0$: α + β = (20)/(12) = (5)/(3)$\alpha + \beta = \frac{20}{12} = \frac{5}{3}$, αβ = (3λ)/(12) = (λ)/(4)$\alpha\beta = \frac{3\lambda}{12} = \frac{\lambda}{4}$.

Core Logic

Square the inequality (1)/(2) ≤ |α - β| ≤ (3)/(2)$\frac{1}{2} \le |\alpha - \beta| \le \frac{3}{2}$:

Step 1: Integer Values of Lambda

Since λ in Z$\lambda \in \mathbb{Z}$, the valid integer values are λ = 1, 2$\lambda = 1, 2$.

Pattern Recognition

Convert root difference inequality into quadratic discriminant bounds for quick integer extraction.

Chapter Mix

Class 11 Maths: Quadratic Equations