Algebra

Vieta's Formula

Vieta's Formula Problem Solving

         

Suppose one of the roots of the quadratic equation x2+6ax+b=0x^2+6ax+b=0 is 3+8.\sqrt{3+\sqrt{8}}. If aa and bb are rational numbers, what is the value of 15ab?15ab?

What is the sum of the two roots of the following equation: (3x1355)2+4(3x1355)+5=0?(3x-1355)^2+4(3x-1355)+5=0?

AA and BB are the two roots of the quadratic equation f(x)=0f(x)=0. If A+B=6A+B=6, what is the sum of the two roots of f(2x21)=0f(2x-21)=0?

Let aa and bb be the roots of the quadratic polynomial x2x5=0.x^2-x-5=0. Let f(x)f(x) be a cubic polynomial such that f(a)=a,f(b)=b,f(a)=a, f(b)=b, f(a+b)=a+b,f(a+b)=a+b, and f(ab)=155.f(ab)=-155. What is the value of f(6)f(6)?

Find the sum of all real solutions to the equation 64x20572x+3072=0.6 \cdot 4^x - 2057 \cdot 2^x + 3072 = 0.

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