heureka

This might be a silly question, but how did you know that that would be the answer?

 \(\begin{array}{rcrcr} \color{blue}3x^2 &-& x^2 &=& \color{red} 2x^2 \\ \color{blue}7x &-& (- 2x) &=& \color{red}9x\\ \color{blue}0 &-& 6 &=& \color{red}-6 \end{array}\)

What Polynomial must be added to X^2-2x+6 so that the sum is 3x^2+7x?

\(\begin{array}{rcrcrcr} && x^2 &-& 2x &+& 6 \\ &\color{red}+& \color{red} 2x^2 &\color{red}+& \color{red}9x &\color{red}-& \color{red}6 \\ \hline &=& \color{blue}3x^2 &\color{blue}+& \color{blue}7x &\color{blue}+& \color{blue}0 \\ \end{array}\)

The average score of the last 3 quizzes of Sofia was 83. The overall average of her 6 quizzes was 86. What was her average score for the first 3 quizzes?

\(\begin{array}{rcl} \frac{s_1+s_2+s_3}{3} &=& ? \\ \hline \frac{s_4+s_5+s_6}{3} &=& 83 \qquad | \qquad \cdot \frac12\\\\ \frac{s_4+s_5+s_6}{6} &=& \frac{83}{2}\\ \hline \frac{s_1+s_2+s_3+s_4+s_5+s_6}{6} &=& 86 \\ \frac{s_1+s_2+s_3}{6} + \frac{s_4+s_5+s_6}{6} &=& 86 \\ \frac{s_1+s_2+s_3}{6} + \frac{83}{2} &=& 86 \qquad | \qquad \cdot 2 \\ \frac{s_1+s_2+s_3}{3} + 83 &=& 86 \cdot 2 \\ \frac{s_1+s_2+s_3}{3} + 83 &=& 172\\ \frac{s_1+s_2+s_3}{3} &=& 172 - 83 \\ \mathbf{\frac{s_1+s_2+s_3}{3}} & \mathbf{=} & \mathbf{89} \\ \end{array}\)

The average score for the first 3 quizzes was 89

normalcdf(-5,-1.414) = 0.07868066459

How do I turn Decimals into Percentages?

\(\begin{array}{rcl} \text{Decimals} \\ &=& \text{Decimals}\cdot \frac{100}{100}\\ &=& \text{Decimals}\cdot 100\cdot \frac{1}{100} \qquad | \qquad \frac{1}{100} = \%\\ &=& \text{Decimals}\cdot 100\cdot \%\\ \end{array}\)

write an equation in point-slope form of the line that passes through given point and with the given slope m. (3, -4); m= 6

\(\begin{array}{rcr} x_1 &=& 3\\ y_1 &=& -4\\ m&=& 6 \end{array}\)

\(\begin{array}{rcl} \text{Line: Point Slope Form}\\ \boxed {~ \frac{ y-y_1} {x-x_1} = m ~} \end{array}\)

\(\begin{array}{rcl} \frac{ y-y_1} {x-x_1} &=& m \\ \frac{ y-(-4)} {x-3} &=& 6 \\ \frac{ y+4} {x-3} &=& 6 \\ y+4 &=& 6\cdot (x-3) \\ y+4 &=& 6\cdot x - 18\\ y &=& 6\cdot x - 18 - 4\\ \mathbf{y} & \mathbf{=} & \mathbf{6\cdot x - 22} \\ \end{array}\)

Naturally occuring bromine (relative atomic mass = 79.9) consists of a mixture of two isotopes: 79Br and  81Br. Calculate the relative abundance of the naturally occurring isotopes of bromine.

maya makes trail mix by combining 1/3 of cup mixed nuts, 1/4 cup of dried fruit, 1/6 cup of chocolate morsels. What is the total amount of ingredients in her trail mix?

\(\begin{array}{rcll} \frac13+\frac14+\frac16 &=& x \qquad & | \qquad \cdot 3\\\\ 3\cdot \frac13+3\cdot \frac14+3\cdot \frac16 &=& 3\cdot x \\\\ 1+ \frac34+\frac12 &=& 3\cdot x \\\\ 1+ \frac34+\frac12 &=& 3\cdot x \qquad & | \qquad \cdot 4\\\\ 4\cdot 1+ 4\cdot \frac34+4\cdot \frac12 &=& 4\cdot 3\cdot x\\\\ 4+ 3 +2 &=& 12\cdot x\\\\ 9 &=& 12\cdot x\\\\ 12\cdot x &=& 9 \\\\ x &=& \frac{9}{12} \qquad & | \qquad :3\\\\ \mathbf{x} & \mathbf{=} & \mathbf{\frac{3}{4}} \\ \mathbf{\frac13+\frac14+\frac16} &\mathbf{=} & \mathbf{\frac{3}{4}} \end{array}\)

Wie kann ich schnell Zahlen von 1 bis 200 zusammenrechnen? Also 1+2+3+4+5+6 usw.

In the formula \(P = \frac{7}{12}\cdot Q +60\), if P is increased by 35, then what is the increase in Q?

\(\begin{array}{rcl} P+35 &=& \frac{7}{12}\cdot (Q+x) +60 \\ P+35 &=& \frac{7}{12}\cdot Q + \frac{7}{12}\cdot x +60 \\ P+35 &=& \underbrace{\frac{7}{12}\cdot Q + 60}_{=P} +\frac{7}{12}\cdot x\\ P+35 &=& P +\frac{7}{12}\cdot x\\ 35 &=& \frac{7}{12}\cdot x\\ \frac{7}{12}\cdot x &=& 35\\ x &=& 35\cdot \frac{12}{7} \\ \mathbf{x} & \mathbf{=} & \mathbf{60} \\ \end{array}\)

the answer is (B)