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Classic complete the square problem ^_^

If you expand the right side, we have

\(x^2 - 3x + 2 = x^2 - 2kx + k^2 + p\).

Because like terms have to match on both sides, we garner \(-3x = -2kx \implies k = \frac{3}{2}\).

Now, we need \(2 = k^2 + p.\) Substituting the value for \(k\) we just found, we get \(2 = \left(\frac{3}{2}\right)^2 + p \implies 2 = \frac{9}{4} \implies p = -1/4\).

So, we can conclude the answer \(d.) -1/4\).

(4x)/ 8  =  y    .......so.....

(4x) / 8  =   3

(1/2)x  = 3

x =   3  / (1/2)  =   6

Formula  for the nth Lucas number  =

Lucas(n) = Phi^n + ( –phi )^n

Where   Phi   =     (1 + sqrt 5)/ 2

And phi  =  ( sqrt 5 - 1 ) / 2

So

L (100)  =    [ (1 + sqrt 5)/2 ]^100 +  [ (-1)( sqrt 5 - 1)/2 ]^100  = 792070839848372253127

3x^2  + 7x + k  = 0

Let  the roots  be  a  and b

So

1/a + 1/b =  7/3

(a + b) / ab  = 7/3

The product of the roots  =   k/3

So

ab =  k/3

So

(a + b)  / (k/3)   =  7/3

3 (a + b)  =  (7/3)k

a + b  =  (7/9)k

And the sum of the roots =   -7/3

So

(-7/3) = (7/9)k

k =  (-7/3) / (7/9)  =  -3

What is sin GIH?  What is sin GHI?

\(sin\ GIH=sin \ arccos\ \frac{9}{10}=sin\ 25.842°=\color{blue}0.43589\\ sin\ GHI=sin\ (180-2\cdot 25.842)°=sin\ 128.316°=\color{blue}0.7846\)

  !

I am not particularly comfortable with questions like this either but I think what I have done is correct.

We can rewrite \((a-4)+(b-4)\) as \(a+b-8\)

Using Vieta's formula, the sum of the roots is \(\large { -b \over a}\). Substituting the values, we find that \(a+b= -{1 \over 7}\)

Substracting 8 from this will yield you the answer...

Substitute it like this: \((7 \times 2) +(3 \times 3)-48\)

 Find u/v + v/u.

Hello Guest!

\(3x^2+5x+7=2x^2-6x+1\\ x^2+11x+6=0\\ x=-\frac{11}{2}\pm \sqrt{\frac{121}{4}-6}\\ \{u,v\}=\{ -5.5+\sqrt{\frac{97}{4}}, -5.5-\sqrt{\frac{97}{4}}\}\)

\(\dfrac{u}{v}=\frac{ -5.5+\sqrt{\frac{97}{4}} }{ -5.5-\sqrt{\frac{97}{4}}}=0.05521\)

\(\dfrac{v}{u}=18.111\)

\({\color{blue}\dfrac{u}{v}+\dfrac{v}{u}=}\frac{ -5.5+\sqrt{\frac{97}{4}} }{ -5.5-\sqrt{\frac{97}{4}}}+\frac{ -5.5-\sqrt{\frac{97}{4}} }{ -5.5+\sqrt{\frac{97}{4}}}=\color{blue}18.1\overline 6=\frac{109}{6}\)

  !

We can rewrite \(\large {{u \over v} + {v \over u}}\) as \(\large{{u^2+v^2} \over {uv}}\). We can also rewrite the numerator as: \(\large {{(u+v)^2-2uv} \over {uv}}\)

Using Vieta's, we know that \(u+v=-11\) and that \(uv=6\)

From here, substitute the values and simplify.

Feel free to ask if you need any help!!!

Using Vieta's, the sum of the roots \((a+b)\) is \({11 \over 5}\)

Plugging this in, we have: \(\large{1 \over {11 \over 5}} = \color{brown}\boxed{5 \over 11}\)

Ms. Ramirez gave a test in her math class and created the histogram shown to
display her students’ scores. If no two students scored the same number of points
on this test, what is the least possible value of the median score?

What is sin GIH?  What is sin GHI?

Hello Guest!

law of cosines:

\(a^2= c^2+b^2-2bc\cdot cos\ \alpha\\ \color{blue}sin\ \alpha=sin\ (arc cos\ \dfrac{c^2+b^2-a^2}{2bc})\)

You can calculate it yourself.

  !

Area of a circle = pi*r^2          (*)

they give you the diameter of the circle so using the relationship btwn diameter, d, and radius, r ---> r = d/2

thus the radius of this particular circle is 15/2.

Now plugging the values into equation (*), A = pi*(15/2)^2 -----> A = 225pi/4

6. from A there are only two paths you can take (A -> C and A -> D) from either C or D there are three paths each and each subsequent choice leads to a node for which there are only possible path remaining. (i.e. A to C to F which means that from F you can only move to D then to E then to B; one path remaining at each of the remaining 3 nodes) 

I will just add the pics here

is is the second linked pic.

First solve the quadratic by using the quadratic formula, x = (-(-11)+/- sqrt((-11)^2-4(5)(4)))/2(5) ---> x = (11+/- sqrt(41))/10. 

sps. a = (11+sqrt(41))/10 and b = (11-sqrt(41))/10

then a+b just equals 22/10 = 11/5 

the question asks for the reciprocal therefore the answer is 5/11.

When you multiply the equation by 36, you get: \(9x+2y=36\)

The x-intercept is 4, because the equation is true when y = 0. 

The y-intercept is 18 by the same logic. 

The slope is \(-18 \over 4 \).

Thus, \(a+b+m=18+4-{18 \over 4} = \text{____}\)

In a class of 27 students, 18 are female and 8 have an A in the class. There are 2 students who are female and have an A in the class. What is the probability that a student chosen randomly from the class has an A?  

Try it this way:  In a class of 27 students, 18 are female and 8 have an A in the class.  

Since the second sentence specifies that 2 females have an A, I conclude that the ambiguous phrase "18 are female and 8 have an A in the class" must mean that it is 8 out of the whole class, not just of the 18 females.  That means the likelihood is 8 out of 27, written 8/27 or 0.296.  

Let

a  = the width of the the top left rectangle

b =  its height

c = the height of the bottom left rectangle

d = the width of the top rght rectangle

Then x  =  dc

ab = 3                                 ab  =  3                  

ac = 7                                 db  =12                                

b : c  =  3  :  7                     a : d  =  3 : 12 =  1 : 4                                         

Let b + c  = 10                   Let  a + d  =  5          

Then c   = 7                       Then d = 4

dc =   4 * 7    = 28  =  x

\( $4x^{1/3}-2\frac{x}{x^{2/3}}=7+\sqrt[3]{x}/2$\)

4x^(1/3) - 2x^(1/3)  =  7  + x^(1/3)  / 2

2x^(1/3)  = 7  + x^(1/3) / 2           multiply through by 2

4x^(1/3)  = 14  + x^(1/3)

3x^(1/3)  =  14

x^(1/3)  = 14 / 3                  cube both sides

x = (14/ 3)^3  =  2744 /  27

If the diameter is 14,  the radius  =  7

Circumference    = 2 * pi * r

Circumference   = 2 * 3.14 *  7  =   ???

Here :

https://web2.0calc.com/questions/coordinates_23153