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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.

Forming Quadratic Equations

Which of the following expresses the quadratic equation whose roots are \(p\) and \(6\)?

A) \( x^2 - 6p x + (6 + p) = 0 \)
B) \( x^2 + 6p x - (6+b) = 0 \)
C) \( x^2 + (6 + p) x - 6p = 0 \)
D) \( x^2 - (6+p) x + 6p = 0 \)

In the above diagram, \(AB\) is a diameter of the circle with radius \(21\) and \(CD\) intersects \(AB\) at point \(P.\) Suppose that \(\lvert \overline{CP} \rvert =8\) and \( \lvert \overline{DP} \rvert=12.\) If the quadratic equation whose two roots are \(\lvert \overline{AP} \rvert\) and \( \lvert \overline{BP} \rvert\) can be expressed as \(x^2+ax+b=0,\) what is the value of \(a+b?\)

Which of the following expresses the quadratic equation whose roots are \(4\) and \(-6\)?

A) \( x^2 + 2 x + 24 = 0 \)
B) \( x^2 - 2 x + 24 = 0 \)
C) \( x^2 + 2 x - 24 = 0 \)
D) \( x^2 - 2 x - 24 = 0 \)

Let \(\alpha\) and \(\beta\) be the two roots of the quadratic equation \[x^2-mx+n=0,\] and \(\alpha+\beta\) and \(\alpha\beta\) be the two roots of the quadratic equation \[x^2-8x+1=0.\] Then what is the value of \(m^3+n^3 ?\)

Let \(\alpha\) and \(\beta\) be the two roots of the quadration \[x^2-2x-2=0.\] If the quadratic equation whose two roots are \(\alpha^3\) and \(\beta^3\) can be expressed as \(x^2+ax+b=0,\) what is the value of \(3ab?\)

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