Let α1, α, be the roots of a x^2+b x+c=0 and β1, β2 the roots of p x^2+q x+r=0 so that the syste

step 1 In problem 1 we have alpha 1 and alpha 2 the roots of the equation a x squared plus bx plus c eq

Let α1, α, be the roots of a x^2+b x+c=0 and β1, β2 the...

Let $\alpha_{1}, \alpha$, be the roots of $a x^{2}+b x+c=0$ and $\beta_{1}, \beta_{2}$ the roots of $p x^{2}+q x+r=0$ so that the system of equations $\alpha_{1} y$ $+\alpha_{2} z=0 ; \beta_{1} y+\beta_{2} z=0$ has a non-zero solution. Prove that $p, q, r$ are in $G P$ if and only if $a, b, c$ are in $G P$

Key Concepts

Geometric Progression of Coefficients

A sequence of numbers is said to be in a geometric progression (GP) if the square of the middle term equals the product of the extreme terms, that is, b² = ac for three numbers a, b, and c. This concept, when applied to the coefficients of quadratic equations, links the algebraic form of the equation with multiplicative relationships among the coefficients. In problems that require proving equivalence conditions between two quadratic equations, establishing that their coefficients are in GP is a key analytical step.

Vieta's Formulas

Vieta's formulas establish a connection between the coefficients of a polynomial and the sums and products of its roots. When dealing with quadratic equations, these formulas allow one to express the sum and product of the roots in terms of the coefficients. Such relationships are essential in proving conditions where the roots' properties mirror the properties of the coefficients, such as when determining the conditions under which the coefficients form a geometric progression.

Non-trivial Solutions of Homogeneous Systems

A homogeneous system of linear equations (where all constant terms are zero) has non-trivial solutions (solutions other than the trivial all-zero solution) if and only if the determinant of its coefficient matrix is zero. This condition is vital because it relates the properties of the system's coefficient matrix directly to the existence of infinitely many solutions, and it is frequently used in proofs involving relationships among parameters or coefficients.

Determinant Condition

The determinant of a square matrix provides a scalar value which encodes certain properties of the matrix, including whether the matrix is invertible. In the context of a homogeneous system of equations, a zero determinant signifies that the system does not have a unique solution, which in turn implies the existence of a non-trivial solution. This concept is used to derive relationships among variables or coefficients in problems involving quadratic equations.

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