Problem

Problem

The quadratic equation ax2−6x+a−2=0ax^2 - 6x + a - 2 = 0 , a≠0a \ne 0 , has one root which is double the other. a. Let the roots be α\alpha and 2α2\alpha . Hence find two equations involving α\alpha . b. Find aa and the two roots of the quadratic equation.
Intuitive explanation

The key insight is to use the relationship between roots and coefficients. If one root is double the other, call them rr and 2r2r—then their sum and product are both determined by aa. Set up Vieta's formulas for the sum and product, use the doubling condition to eliminate one variable, and solve for aa. This turns an abstract constraint into a concrete algebraic system.

Intuitive explanation

The key insight is to use the relationship between roots and coefficients. If one root is double the other, call them rr and 2r2r—then their sum and product are both determined by aa. Set up Vieta's formulas for the sum and product, use the doubling condition to eliminate one variable, and solve for aa. This turns an abstract constraint into a concrete algebraic system.

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