heureka

for the equation 5x^3-7x^2+4x-2, the answer is 1/5 + 3/5i, where exactly does the i come from?

The zero set of discriminant of the cubic  \({\displaystyle ax^{3}+bx^{2}+cx+d\,} \) has discriminant \({\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}\)
The discriminant is zero if and only if at least two roots are equal.

If the coefficients are real numbers, and the discriminant is not zero,
the discriminant is positive if the roots are three distinct real numbers,

and negative if there is one real root and two complex conjugate roots.

\(\begin{array}{|rcll|} \hline && 5x^3-7x^2+4x-2 \\ && ax^{3}+bx^{2}+cx+d \\ &&{} a = 5, \quad b=-7, \quad c = 4, \quad d = -1 \\\\ && b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd \\ &=& (-7)^{2}\cdot 4^{2}-4\cdot 5\cdot 4^{3}-4(-7)^{3}\cdot (-1)-27\cdot 5^{2}\cdot (-1)^{2} +18\cdot 5\cdot (-7) \cdot 4 \cdot (-1) \\ &=& -23 \\ \hline \end{array}\)

There is one real root and two complex conjugate roots:

\(x_1 = 1, \quad x_2 = \frac15 - \frac{3 i}{5}, \quad x_3 = \frac15 + \frac{3 i}{5} \)

In making a topographical map, it is not practical to measure directly heights of structures such as mountains.

This exercise illustrates how some such measurements are taken.

A surveyor whose eye is a = 6 feet above the ground views a mountain peak that is c = 5 horizontal miles distant.

(See the figure below.)

Directly in his line of sight is the top of a surveying pole that is 10 horizontal feet distant and b = 7 feet high.

How tall is the mountain peak?

Note: One mile is 5280 feet.

_________ft

\(\begin{array}{|rcll|} \hline \dfrac{H-a}{c} &=& \dfrac{b-a}{10} \\ H-a &=& c\cdot \left( \dfrac{b-a}{10} \right) \\ H &=&a+ c\cdot \left( \dfrac{b-a}{10} \right) \quad & | \quad a = 6\ feet, \quad b = 7\ feet, \quad c = 5\ miles\cdot \frac{5280\ feet}{mile} \\\\ H &=&6+ 5\cdot 5280 \left( \dfrac{7-6}{10} \right) \\\\ H &=&6+ 5\cdot 528 \\\\ H &=&2646\ feet \\ \hline \end{array} \)

the mountain peak is 2646 feet tall.

what is the value of the discriminant?

Degree 2
The quadratic polynomial \({\displaystyle ax^{2}+bx+c\,} \,\) has discriminant \({\displaystyle b^{2}-4ac\,.}\)

The square root of the discriminant appears in the quadratic formula for the roots of the quadratic polynomial:

 \( {\displaystyle x_{1,2}={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}\)

The discriminant is zero if and only if the two roots are equal.

If a, b, c are real numbers, the polynomial has two distinct real roots if the discriminant is positive,

and two complex conjugate roots if it is negative.

Degree 3
The zero set of discriminant of the cubic  \( {\displaystyle ax^{3}+bx^{2}+cx+d\,}\)  has discriminant

  \( {\displaystyle b^{2}c^{2}-4ac^{3}-4b^{3}d-27a^{2}d^{2}+18abcd\,.}\)
The discriminant is zero if and only if at least two roots are equal.

If the coefficients are real numbers, and the discriminant is not zero, the discriminant is positive if the roots are three distinct real numbers,

and negative if there is one real root and two complex conjugate roots.

Degree 4
The discriminant of the quartic polynomial  \( {\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e\,} \)has discriminant
\({\displaystyle {\begin{aligned}{}&256a^{3}e^{3}-192a^{2}bde^{2}-128a^{2}c^{2}e^{2}+144a^{2}cd^{2}e\\ &{}-27a^{2}d^{4}+144ab^{2}ce^{2}-6ab^{2}d^{2}e-80abc^{2}de\\ &{}+18abcd^{3}+16ac^{4}e-4ac^{3}d^{2}-27b^{4}e^{2}+18b^{3}cde\\ &{}-4b^{3}d^{3}-4b^{2}c^{3}e+b^{2}c^{2}d^{2}\,.\end{aligned}}}\)
The discriminant is zero if and only if at least two roots are equal.
If the coefficients are real numbers and the discriminant is non-zero,
the discriminant is negative if there is two real root and two complex conjugate roots,
and it is positive if the roots are either all real or all non-real.

whats the nth term of 2,8,18,32

\(\small{ \begin{array}{lrrrrrrrrrr} & {\color{red}d_0 = 2} && 8 && 18 && 32 && 50 && \cdots \\ \text{1. Difference } && {\color{red}d_1 = 6} && 10 && 14 && 18 && \cdots \\ \text{2. Difference } &&& {\color{red}d_2 = 4} && 4 && 4 && \cdots \\ \end{array} }\)

\(\boxed{~ \begin{array}{rcl} a_n &=& \dbinom{n-1}{0}\cdot {\color{red}d_0 } + \dbinom{n-1}{1}\cdot {\color{red}d_1 } + \dbinom{n-1}{2}\cdot {\color{red}d_2 } \end{array} ~}\)

\(\begin{array}{|rcll|} \hline a_n &=& \dbinom{n-1}{0}\cdot {\color{red} 2 } + \dbinom{n-1}{1}\cdot {\color{red} 6 } + \dbinom{n-1}{2}\cdot {\color{red} 4 } \\\\ && \binom{n-1}{0} = 1 \\ && \binom{n-1}{1} = n-1 \\ && \binom{n-1}{2} = \left( \frac{n-1}{2} \right) \cdot \left( \frac{n-2}{1} \right)\\\\ a_n &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + \left( \frac{n-1}{2} \right) \cdot (n-2) \cdot {\color{red} 4 } \\ &=& {\color{red} 2 } + (n-1)\cdot {\color{red} 6 } + (n-1) \cdot (n-2) \cdot {\color{red} 2 } \\ &=& 2 + 6n-6 + (2n-2) \cdot (n-2) \\ &=& -4 + 6n + 2n^2-4n-2n+4 \\ \mathbf{a_n} & \mathbf{=} & \mathbf{2n^2} \\ \hline \end{array}\)

Example:

\(\begin{array}{|rclcl|} \hline a_1 &=& 2\cdot 1^2 &=& 2 \\ a_2 &=& 2\cdot 2^2 &=& 8 \\ a_3 &=& 2\cdot 3^2 &=& 18 \\ a_4 &=& 2\cdot 4^2 &=& 32 \\ a_5 &=& 2\cdot 5^2 &=& 50 \\ \dots \\ \mathbf{a_n} &\mathbf{=}& \mathbf{2\cdot n^2} \\ \hline \end{array}\)

1. What is the area of this figure?

To check the area you can use the Pick's theorem:

Given a simple polygon constructed on a grid of equal-distanced points (i.e., points with integer coordinates) such that all the polygon's vertices are grid points, Pick's theorem provides a simple formula for calculating the area A of this polygon in terms of the number i of lattice points in the interior located in the polygon and the number b of lattice points on the boundary placed on the polygon's perimeter:

\({\displaystyle A=i+{\frac {b}{2}}-1.}\)

Example:

\(\begin{array}{|rcll|} \hline i &=& 45 \\ b &=& 20 \\ A &=& 45 + \frac{20}{2} -1 \\ \mathbf{A} &=& \mathbf{54} \\ \hline \end{array}\)

was ist die wurzel von pi

\( {\displaystyle \Gamma ({\tfrac {1}{2}})={\sqrt {\pi }}} = 1,77245 38509 05516 02729 …\)

a box contains
5 purple marbles,
3 green marbles and
2 orange marbles.
two consecutive draws are made from the box withous replacement of the first draw
what is the probability that the first marble is purple and the second is any color except purple.

\(\begin{array}{|rcll|} \hline && \frac{5}{10} \cdot \frac{3}{9}+\frac{5}{10} \cdot \frac{2}{9} \\ &=& \frac{5}{10} \cdot \left( \frac{3}{9}+ \frac{2}{9} \right) \\ &=& \frac{5}{10} \cdot \frac{5}{9} \\ &=& \frac{25}{90} \\ &=& 0.2\bar{7} \quad (27.\bar{7}\ \%) \\ \hline \end{array} \)

The height (in meters) of a shot cannonball follows a trajectory given by
h(t) = -4.9t^2 +14t -0.4 at time t (in seconds).
As an improper fraction, for how long is the cannonball above a height of 6 meters?

\(\begin{array}{|rcll|} \hline h(t) = -4.9t^2 +14t -0.4 & \ge & 6 \\\\ -4.9t^2 +14t -0.4 & = & 6 \\ -4.9t^2 +14t -0.4 -6 & = & 0 \\ -4.9t^2 +14t -6.4 & = & 0 \\ t &=& \frac{-14\pm \sqrt{ 14^2-4\cdot (-4.9)\cdot (-6.4) }}{2\cdot(-4.9)} \\ t &=& \frac{-14\pm \sqrt{ 196-125.44 }}{-9.8} \\ t &=& \frac{-14\pm \sqrt{ 70.56 }}{-9.8} \\ t &=& \frac{-14\pm 8.4 }{-9.8} \\ t &=& \frac{14\pm 8.4 }{9.8} \\\\ \Delta t &=& t_2-t_1 \\ \Delta t &=& \frac{14 + 8.4 }{9.8}-\left(\frac{14- 8.4 }{9.8} \right) \\ \Delta t &=& \frac{14 + 8.4 -14+8.4}{9.8} \\ \Delta t &=& \frac{2\cdot 8.4}{9.8} \\ \Delta t &=& \frac{8.4}{4.9} \\ \Delta t &=& \frac{84}{49} \\ \Delta t &=& \frac{7\cdot 12}{7\cdot 7} \\ \mathbf{ \Delta t } & \mathbf{=} & \mathbf{\frac{ 12}{ 7}} \\ \hline \end{array}\)

The cannonball is \(\frac{12}{7}\) seconds above a height of 6 meters

Wie gross ist die Abwiklung eines 10mx10mx10m Würfels?

Die Abwicklung eines Würfels mit 10m Kantenlänge:

Die Abwiklung eines 10mx10mx10m Würfels ist 600 m²

square roots of -100

\(\begin{array}{|rcll|} \hline && \sqrt{-100} \\ &=& \sqrt{ 100\cdot (-1)} \\ &=& 10 \cdot \sqrt{ -1 } \\ &=& 10 \cdot \sqrt{ -1 } \quad & | \quad \sqrt{z} = \sqrt{-1+0\cdot i} = \sqrt{-1} = \pm i \\ &=&10 \cdot (\pm i) \\ &=&\pm i\cdot 10 \\ \hline \end{array} \)