The roots of the quadratic equation 3x² - px + q = 0 are 10th and 11th terms of an arithmetic progre

Q: Why do we use the sum formula $S_{11} = \frac{11}{2}[2a + 10d]$?

The problem gives us the sum of the first 11 terms, which is 88. This formula is the direct bridge to finding the first term $a$ when we already know the common difference $d$.

Q: Can I use the quadratic formula to find the roots?

You could, but that would require knowing $p$ and $q$ first, which is exactly what we are trying to find! It is much more efficient to find the roots from the AP properties first.

Q: Is there a shortcut for $q - 2p$?

While there might be algebraic manipulations, the most reliable path is to find $p$ and $q$ individually using Vieta's formulas. It ensures accuracy and avoids potential sign errors.

Question

The roots of the quadratic equation $3x^{2}-px+q=0$ are $10^{th}$ and $11^{th}$ terms of an arithmetic progression with common difference $\frac{3}{2}$. If the sum of the first 11 terms of this arithmetic progression is 88, then $q-2p$ is equal to

Sigma Percentile · Accepted Answer

### Analyzing the Arithmetic Progression

We begin by decoding the properties of the Arithmetic Progression (AP). We are given the common difference $d = \frac{3}{2}$ and the sum of the first 11 terms $S_{11} = 88$.

Using the sum formula for an AP, $S_{n} = \frac{n}{2}[2a + (n-1)d]$, we substitute the known values:

$$88 = \frac{11}{2}[2a + 10d]$$

Dividing both sides by 11 and multiplying by 2, we simplify the expression:

$$16 = 2a + 10d \Rightarrow a + 5d = 8$$

Substituting $d = \frac{3}{2}$ into the equation, we find the first term $a$:

$$a + 5\left(\frac{3}{2}ight) = 8 \Rightarrow a + 7.5 = 8 \Rightarrow a = 0.5 = \frac{1}{2}$$

### The Bridge: Finding the Roots

With the first term $a = \frac{1}{2}$ and common difference $d = \frac{3}{2}$ identified, we calculate the 10th and 11th terms, which serve as the roots $\alpha$ and $\beta$ of the quadratic equation $3x^{2}-px+q=0$.

The 10th term is calculated as:

$$T_{10} = a + 9d = \frac{1}{2} + 9\left(\frac{3}{2}ight) = \frac{1}{2} + \frac{27}{2} = 14$$

The 11th term is calculated as:

$$T_{11} = a + 10d = \frac{1}{2} + 10\left(\frac{3}{2}ight) = \frac{1}{2} + \frac{30}{2} = 15.5$$

Thus, our roots are $\alpha = 14$ and $\beta = 15.5$.

### The Vieta's Masterclass

We now apply Vieta's formulas to the quadratic equation $3x^{2}-px+q=0$ to determine the coefficients $p$ and $q$. The sum of the roots is given by:

$$\alpha + \beta = \frac{p}{3}$$

Substituting our values:

$$14 + 15.5 = 29.5 = \frac{59}{2} = \frac{p}{3} \Rightarrow p = \frac{177}{2} = 88.5$$

The product of the roots is given by:

$$\alpha \beta = \frac{q}{3}$$

Substituting our values:

$$14 	imes 15.5 = 217 = \frac{q}{3} \Rightarrow q = 651$$

### Final Calculation

To reach the final result, we compute the value of $q - 2p$:

$$q - 2p = 651 - 2(88.5) = 651 - 177 = 474$$

The final answer is **474**.

Enter Numerical Value:

Visualized Solution

Problem Overview

Arithmetic Progression Data

Applying the Sum Formula

Simplifying the Equation

Calculating the First Term $a$

Calculating the $1 0^{t h}$ Term

Calculating the $1 1^{t h}$ Term

Sum of Roots

Product of Roots

Final Calculation

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Arithmetic Progression

The Bridge

The Vieta's Masterclass

Final Calculation

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Enter Numerical Value:

Visualized Solution

Problem Overview

Arithmetic Progression Data

Applying the Sum Formula

Simplifying the Equation

Calculating the First Term .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a

Calculating the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 10th Term

Calculating the .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } 11th Term

Sum of Roots

Product of Roots

Final Calculation

The Sigma Insight: Arithmetic Progression (A.P.)

Analyzing the Arithmetic Progression

The Bridge

The Vieta's Masterclass

Final Calculation

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