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Vieta's Formula

Vieta's formula relates the coefficients of polynomials to the sum and products of their roots. This can provide a shortcut to finding solutions in more complicated algebraic polynomials.

Problem Solving - Basic

         

Suppose one of the roots of the quadratic equation \[x^2+6ax+b=0\] is \(\sqrt{3+\sqrt{8}}.\) If \(a\) and \(b\) are rational numbers, what is the value of \(15ab?\)

What is the sum of the two roots of the following equation: \[(3x-1355)^2+4(3x-1355)+5=0?\]

\(A\) and \(B\) are the two roots of the quadratic equation \(f(x)=0\). If \(A+B=6\), what is the sum of the two roots of \(f(2x-21)=0\)?

Let \(a\) and \(b\) be the roots of the quadratic polynomial \(x^2-x-5=0.\) Let \(f(x)\) be a cubic polynomial such that \(f(a)=a, f(b)=b,\) \(f(a+b)=a+b,\) and \(f(ab)=-155.\) What is the value of \(f(6)\)?

Find the sum of all real solutions to the equation \[6 \cdot 4^x - 2057 \cdot 2^x + 3072 = 0.\]

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