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1610 j

Don't you know how to add fractions? They have the same denominators, so \(\frac{5}{6}+\frac{4}{6}=\frac{9}{6}=\frac{3}{2}\)

Express k as a common fraction.

Hello Guest!

\(2x^2- 5x + k = x^2 - 4x\ |\ -(x^2-4x) \\ x^2-x+k=0\\ p-q\ formula\\ x=0.5\pm \sqrt{0.25-k}\\ 0.25-k=0\\ \color{blue}k=0.25=\frac{1}{4}\\ \color{blue}x=0.5\)

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Hi Guest!

Rewrite the first two equations as follows:

\(xy=15+z\)  (1)

\(xz=15+y \)  (2)

Subtracting (2) from (1) gives:

\(x(y-z)=-(y-z)\)       (Notice: \(z-y=-(y-z)\) ).

\(x(y-z)+(y-z)=0\)        (Add (y-z) to both sides to make R.H.S = 0 , and factor (y-z)):

\((y-z)(x+1)=0\)

Thus, \(y=z, \text{or } x=-1 (rejected)\)  (As given x,y,z > 0).

Therefore, \(y=z\) only.

Also, using the second and third equation:

\(xz=15+y\)  (2)

\(yz=15+x\)  (3)

Subtracting (3) from (2) to get: 

\(z(x-y)=-(x-y)\)

\(z(x-y)+(x-y)=0\)

\((x-y)(z+1)=0\)

Therefore, \(x=y\)        (z>0 so z=-1 is rejected).

But we found that:  \(y=z\)

Thus, It must mean that they are all equal to each other. That is,  \(x=y=z\)

So our system of equations becomes:

\(x^2-x=15\) \(\iff x^2-x-15=0 \iff x=\frac{1+\sqrt{61}}{2}\)  (A quadratic equation in x, which can be solved by the quadratic formula.)

So the desired answer is:

\(xyz=x^3=\frac{(1+\sqrt{61})^3}{8}=23+8\sqrt{61}\)

I hope this helps!

Let the weight of the empty tank ==E

Let the weight of fuel when tank is full ==F

E  +  0.25F ==2,500.....................(1)

E  +  0.75F ==3,100......................(2), solve for E, F

Subtract (1) from (2), and we have:

0.50F ==600

F ==600 / 0.50 ==1,200 tons - weight of fuel when tank is full

Sub this into (1) above:

E + [0.25 * 1200] ==2,500

E  +   300 == 2,500

E ==2,500 - 300 ==2,200 tons - weight of the empty tank.

A fuel tanker weighs 2500 tons when 25% full and 3100 tons when 75% full. What is its weight in tons when empty? 

I'll make a little table so it's easy to visualize.  

% Full       Weight (tons)  

       0            2200  

     25            2500  

     50            2800  

     75            3100 

   100            3400     (Values in maroon are interpolated from the given data.)

From 25% to 75% the weight goes from 2500 to 3100   

The change in fullness is 50% and the change in weight is 600 

Half of 50 is 25 and half of 600 is 300 

So every 25% of the tank changes the weight by 300 tons 

We subtract those 300 tons from the weight when 25% full, thus:   2500 – 300 = 2200 

Therefore, the weight of the empty tanker is 2200 tons  

_.

The right side of the equation is equal to 1, which means y=0.

< will the 13th number in the pattern be even or odd? >    

The 13th number in the series will be even.  

So will the 14th, the 15th, the 16th, and every number thereafter.

This is an arithmetic series, where you add six to the previous number.

When you add an even number to an even number, the sum is even. 

13^5*13^3*13^8 = 13^17 * 13^ -1

To solve this problem, use vertex form: \(y = a(x-h)^2 +k\), where the vertex occurs at \((h,k)\)

Substituting the vertex, we have: \(y = a(x-2)^2 - 1\).

Now, plugging in the point \((3,1)\) and solving for a tells us that \(a = 2\). 

This means that the quadratic in vertex form is \(2(x-2)^2 - 1 = \color{brown}\boxed{2x^2 - 8x + 7}\)

We can write the equation as z^3 = -8.  In exponential form, -8 = 8*e^(pi*i).

Then using roots of unity, the solutions are -2, 1 - i*sqrt(5), 1 + i*sqrt(5).

(y+z)/z = 7/3.

thx and nicely done.

The triangle could be 668, 669, 670, in which case the difference is \(\color{brown}\boxed{1}\)

look here at omi's answer: https://web2.0calc.com/questions/2-in-one-help-please

Rectangle area \(= l \times w  =   4 \times 2 = 8^2\)

CPhill's awesome answer:

a^3  +  3ab^2  =  679

3a^2b  + b^3   =  -652

Note  that   ( a + b)^3  =  ( a^3  + 3a^2b) + (3ab^2  + b^2) =  (a^3 + 3ab^2 )  + ( 3a^2b + b^3)

So

( a + b)^3  =   679  - 652

(a + b)^3  =  27         take the cube root of both sides

(a + b)  =  3

 Answer is 1/32

numbers through 1 and 8 that can add up to 15: 7 and 8 OR 8 and 7, meaning there are 2 possibilities for a sum of 15

8*8=64 total possibilities

yeah i didn't understand the problem either 

Looks right.

\(\phantom{Express your answer in terms of pi}\)

The two externally tangent circles each have a radius of 1 unit. Each circle is tangent to three sides of the rectangle. What is the area of the shaded region?    

The long side (length) of the rectangle is 4 radii long, i.e., 4 units 

The short side (width) of the rectangle is 2 radii long, i.e., 2 units 

Area of the rectangle is length times width = 4 • 2 = 8 units²  

_.

The third piece of info tells us that f(x) is parallel to g(x). 

Does this clear things up?

The smallest positive integer n ==95,832

198 x 95,832 ==18,974,736^(1/4) == 66

Convert the second equation into a quadratic: \(y = x^2 - 3\)

Substitute it into the first equation: \(-x^2 + x + 3 = 3\)

Simplify: \(-x^2 + x = 0\)

Note that because the discriminant (\(1 - (4 \times 1 \times 0) = 1 \)) is positive, there will be exactly \(\color{brown}\boxed{2}\) real solutions. 

\(6^{3t - 1} = 36^t\)

\(6^{3t-1} = {6^2}^t\)

\(6^{3t-1} = 6^{2t}\)

\(3t - 1 = 2t\)

Can you take it from here?