Polynomial systems, quadratic-with-parameter, advanced word problems.
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Q1 · Advanced Algebra · MEDIUM
If one root of the quadratic equation x^2 - (k+2)x + k^2 = 0 is twice the other root, then find the value of k.
Q2 · Advanced Algebra · EASY
The system of equations 2x + 3y = 7 and (k-1)x + (k+2)y = 3k has infinitely many solutions. Find the value of k.
Q3 · Advanced Algebra · MEDIUM
A man travels 600 km partly by train and partly by car. If he covers 400 km by train and the rest by car, it takes him 6 hours 30 minutes. If he travels 200 km by train and the rest by car, he takes 30 minutes longer. Find the speed of the train in km/h.
Q4 · Advanced Algebra · HARD
If α and β are the roots of the equation x^2 - px + q = 0, then find the equation whose roots are (α^2 + β^2) and (α^3 + β^3).
Q5 · Advanced Algebra · MEDIUM
If x² + 1/x² = 47, what is the value of x³ + 1/x³?