If $a$ and $b$ are the roots of $x^2 - 3x + p = 0$ and $c$, $d$ are roots of $x^2 - 12x + q = 0$, where $a$, $b$, $c$, $d$ form a G.P. Prove that $(q + p) : (q - p) = 17 : 15$.

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> [!TLDR]
> When four numbers in G.P. are roots of two quadratics, their common ratio and the relationship between sum/product of roots forces a unique ratio between the constants. We find $(q + p) : (q - p) = 17 : 15$.

The heart of this problem lies in connecting two pieces of algebra: Vieta's formulas (which relate roots to coefficients) and the structure of a geometric progression. When four terms form a G.P., each term is the previous one multiplied by a common ratio $r$. This rigid structure, combined with the constraints from the two quadratics, will lock down the relationship between $p$ and $q$.

Let the four terms of the G.P. be $a$, $ar$, $ar^2$, $ar^3$ where $r$ is the common ratio. We need to assign these to the roots of our two equations.

**Setting up the correspondence**

Since $a$ and $b$ are roots of the first equation and $c$, $d$ are roots of the second, and all four form a G.P., we have:
- $a, b, c, d$ in G.P.

The natural assignment is $a = a$, $b = ar$, $c = ar^2$, $d = ar^3$.

**Applying Vieta's formulas**

1. **For the first quadratic** $x^2 - 3x + p = 0$ with roots $a$ and $ar$:
   
   Sum of roots: $a + ar = 3$, so $a(1 + r) = 3$.
   
   Product of roots: $a \cdot ar = p$, so $a^2 r = p$.

2. **For the second quadratic** $x^2 - 12x + q = 0$ with roots $ar^2$ and $ar^3$:
   
   Sum of roots: $ar^2 + ar^3 = 12$, so $ar^2(1 + r) = 12$.
   
   Product of roots: $ar^2 \cdot ar^3 = q$, so $a^2 r^5 = q$.

**Finding the common ratio** …

Property	Formula	Condition
Common ratio	$r = \dfrac{T_{n+1}}{T_n}$	Always
$n$ -th term	$T_n = a r^{n-1}$	Always

Q.If $a$ and $b$ are the roots of $x^2 - 3x + p = 0$ and $c$ , $d$ are roots of $x^2 - 12x + q = 0$ , where $a$ , $b$ , $c$ , $d$ form a G.P. Prove that $(q + p) : (q - p) = 17 : 15$ .

Concept understanding — Geometric Progression

Geometric Progression: The Idea of Repeated Multiplication

Precise Definition

The $n$ -th Term

Sum of $n$ Terms

Sum of an Infinite GP

Quick Reference Table

Q.If aaa and bbb are the roots of x2−3x+p=0x^2 - 3x + p = 0x2−3x+p=0 and ccc, ddd are roots of x2−12x+q=0x^2 - 12x + q = 0x2−12x+q=0, where aaa, bbb, ccc, ddd form a G.P. Prove that (q+p):(q−p)=17:15(q + p) : (q - p) = 17 : 15(q+p):(q−p)=17:15.

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Q.If $a$ and $b$ are the roots of $x^2 - 3x + p = 0$ and $c$ , $d$ are roots of $x^2 - 12x + q = 0$ , where $a$ , $b$ , $c$ , $d$ form a G.P. Prove that $(q + p) : (q - p) = 17 : 15$ .

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