asinus

Find a+b+c. 

Hello Guest!

The functional equation of the parabola shown is:

\(f(x)=(x-1)^2-4=x^2-2x-3\)

\(a=1\\b=-2\\ c=-3\\ \color{blue}a+b+c=-4\)

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Find the minimum possible value of A + B.

Hello Guest!

\(\frac{3}{\sqrt[3]{3} + \sqrt[3]{81}}= \frac{3}{\sqrt[3]{3} +\sqrt[3]{3}\cdot \sqrt[3]{27}}=\frac{3}{\sqrt[3]{3} (1+3)}\cdot \frac{\sqrt[3]{3}\cdot \sqrt[3]{3}}{\sqrt[3]{3}\cdot \sqrt[3]{3}} =\frac{3\cdot \sqrt[3]{9}}{4\cdot 3}=\color{blue}\frac{ \sqrt[3]{9}}{4}\)

\(A+B=9+4=13\)

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 What is this value of k? 

Hello Guest!

   x +  y   + 3z  = 10   | *5

 -4x + 8y + 5z  = 7     | *3

 kx           + 2z = 3

   5x +5y +15z = 50

-12x+24y+15z = 21

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 17x -19y         = 29      | *2

   x +  y   + 3z      = 10   | *2

  kx         + 2z      =  3    | *3

  2x + 2y  +6z      = 20

 3kx         +6z      = 9

 --------------------------

    (2 - 3k)x +   2y = 11     | *19

(38 - 57k)x + 38y =  209

34x             - 38y =   58

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(38 - 23k)x           = 267   | :23

(1.652 - k)x          =11.609

\(x=\dfrac{11.609}{1.652-k}\)

\(y=\frac{34x-58}{38}=\frac{\frac{34\cdot 11.609}{1.652-k}-58}{38} \)

\(y=\dfrac{10.387}{1.652-k}-1.526\)

\(z=\frac{3-kx}{2}=\frac{ 3-\frac{11.609k}{1.652-k} }{2}\)

\(z=1.5-\dfrac{5.8045k}{1.652-k}\)

\(\large k\in \mathbb R\) | -\(\infty\) < k < \(\infty\)

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If f(x) = -2, find the sum of all possible values of x.

Hello Guest!

The sum of all possible values of x is: x = -2 and x = 4.75 

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 Find sin B.

Hello Guest!

\(sin\ B=\frac{\sqrt{9^2-8^2}}{9}=\frac{\sqrt{17 }}{9}=\color{blue}0.45812\)

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I read your question here in English.

Which integer is not in the range of f(x)?

Hello Guest!

-5 is not in the range of f(x).

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If a and b are integers such that a - b = 10, find the minimum value of a*b.

Hello Guest!!

\(a-b=10\\ a=10+b\\ ab=10b+b^2\\ \frac{d(ab)}{db}=10+2b=0\\ b=-5\\ a=5\\ \color{blue}ab_{min}=-25\)

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What is the value of m+b?

Hello guest!

\(P_1(-3,5),\ P_2(0,-14)\\ {\color{blue}m=}\frac{y_2-y_1}{x_2-x_1}=\color{blue} -\frac{19}{3}\\ f(x)=m(x+3)+5=-\frac{19}{3}x-19+5\)

\(f(x)=-\frac{19}{3}x-14\)

\(m+b=-30\frac{1}{3}\)

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