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Algebra

Vieta's Formula - Forming Quadratics

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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