![16{ x }^{ 4 }-41{ x }^{ 2 }+25=0](https://tex.z-dn.net/?f=16%7B%20x%20%7D%5E%7B%204%20%7D-41%7B%20x%20%7D%5E%7B%202%20%7D%2B25%3D0)
![{ x }^{ 4 }={ ({ x }^{ 2 }) }^{ 2 }\\ \\ 16{ ({ x }^{ 2 }) }^{ 2 }-41{ x }^{ 2 }+25=0](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%204%20%7D%3D%7B%20%28%7B%20x%20%7D%5E%7B%202%20%7D%29%20%7D%5E%7B%202%20%7D%5C%5C%20%5C%5C%2016%7B%20%28%7B%20x%20%7D%5E%7B%202%20%7D%29%20%7D%5E%7B%202%20%7D-41%7B%20x%20%7D%5E%7B%202%20%7D%2B25%3D0)
First of all to make our equation simpler, we'll equal
![x^{2}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20)
to a variable like 'a'.
So,
![{ x }^{ 2 }=a](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%202%20%7D%3Da)
Now let's plug
![x^{2}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%20)
's value (a) into the equation.
![16{ ({ x }^{ 2 }) }^{ 2 }-41{ x }^{ 2 }+25=0\\ \\ { x }^{ 2 }=a\\ \\ 16{ (a) }^{ 2 }-41{ a }+25=0](https://tex.z-dn.net/?f=16%7B%20%28%7B%20x%20%7D%5E%7B%202%20%7D%29%20%7D%5E%7B%202%20%7D-41%7B%20x%20%7D%5E%7B%202%20%7D%2B25%3D0%5C%5C%20%5C%5C%20%7B%20x%20%7D%5E%7B%202%20%7D%3Da%5C%5C%20%5C%5C%2016%7B%20%28a%29%20%7D%5E%7B%202%20%7D-41%7B%20a%20%7D%2B25%3D0)
Now we turned our equation into a quadratic equation.
(The variable 'a' will have a solution set of two solutions, but 'x' , which is the variable of our first equation will have a solution set of four solutions since it is a quartic equation (<span>fourth-degree <span>equation) )
Let's solve for a.
The formula used to solve quadratic equations ;
![\frac { -b\pm \sqrt { { b }^{ 2 }-4\cdot t\cdot c } }{ 2\cdot t }](https://tex.z-dn.net/?f=%5Cfrac%20%7B%20-b%5Cpm%20%5Csqrt%20%7B%20%7B%20b%20%7D%5E%7B%202%20%7D-4%5Ccdot%20t%5Ccdot%20c%20%7D%20%20%7D%7B%202%5Ccdot%20t%20%7D%20)
The formula is used in an equation formed like this :
</span></span>
![t{ x }^{ 2 }+bx+c=0](https://tex.z-dn.net/?f=t%7B%20x%20%7D%5E%7B%202%20%7D%2Bbx%2Bc%3D0)
In our equation,
t=16 , b=-41 and c=25
Let's plug the values in the formula to solve.
![t=16\quad b=-41\quad c=25\\ \\ \frac { -(-41)\pm \sqrt { -(41)^{ 2 }-4\cdot 16\cdot 25 } }{ 2\cdot 16 } \\ \\ \frac { 41\pm \sqrt { 1681-1600 } }{ 32 } \\ \\ \frac { 41\pm \sqrt { 81 } }{ 32 } \\ \\ \frac { 41\pm 9 }{ 32 }](https://tex.z-dn.net/?f=t%3D16%5Cquad%20b%3D-41%5Cquad%20c%3D25%5C%5C%20%5C%5C%20%5Cfrac%20%7B%20-%28-41%29%5Cpm%20%5Csqrt%20%7B%20-%2841%29%5E%7B%202%20%7D-4%5Ccdot%2016%5Ccdot%2025%20%7D%20%20%7D%7B%202%5Ccdot%2016%20%7D%20%5C%5C%20%5C%5C%20%5Cfrac%20%7B%2041%5Cpm%20%5Csqrt%20%7B%201681-1600%20%7D%20%20%7D%7B%2032%20%7D%20%5C%5C%20%5C%5C%20%5Cfrac%20%7B%2041%5Cpm%20%5Csqrt%20%7B%2081%20%7D%20%20%7D%7B%2032%20%7D%20%5C%5C%20%5C%5C%20%5Cfrac%20%7B%2041%5Cpm%209%20%7D%7B%2032%20%7D%20)
So the solution set :
![\frac { 41+9 }{ 32 } =\frac { 50 }{ 32 } \\ \\ \frac { 41-9 }{ 32 } =\frac { 32 }{ 32 } =1\\ \\ a\quad =\quad \left\{ \frac { 50 }{ 32 } ,\quad 1 \right\}](https://tex.z-dn.net/?f=%5Cfrac%20%7B%2041%2B9%20%7D%7B%2032%20%7D%20%3D%5Cfrac%20%7B%2050%20%7D%7B%2032%20%7D%20%5C%5C%20%5C%5C%20%5Cfrac%20%7B%2041-9%20%7D%7B%2032%20%7D%20%3D%5Cfrac%20%7B%2032%20%7D%7B%2032%20%7D%20%3D1%5C%5C%20%5C%5C%20a%5Cquad%20%3D%5Cquad%20%5Cleft%5C%7B%20%5Cfrac%20%7B%2050%20%7D%7B%2032%20%7D%20%2C%5Cquad%201%20%5Cright%5C%7D%20)
We found a's value.
Remember,
![{ x }^{ 2 }=a](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%202%20%7D%3Da)
So after we found a's solution set, that means.
![{ x }^{ 2 }=\frac { 50 }{ 32 }](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%202%20%7D%3D%5Cfrac%20%7B%2050%20%7D%7B%2032%20%7D%20)
and
![{ x }^{ 2 }=1](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%202%20%7D%3D1)
We'll also solve this equations to find x's solution set :)
![{ x }^{ 2 }=\frac { 50 }{ 32 } \\ \\ \frac { 50 }{ 32 } =\frac { 25 }{ 16 } \\ \\ { x }^{ 2 }=\frac { 25 }{ 16 } \\ \\ \sqrt { { x }^{ 2 } } =\sqrt { \frac { 25 }{ 16 } } \\ \\ x=\quad \pm \frac { 5 }{ 4 }](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%202%20%7D%3D%5Cfrac%20%7B%2050%20%7D%7B%2032%20%7D%20%5C%5C%20%5C%5C%20%5Cfrac%20%7B%2050%20%7D%7B%2032%20%7D%20%3D%5Cfrac%20%7B%2025%20%7D%7B%2016%20%7D%20%5C%5C%20%5C%5C%20%7B%20x%20%7D%5E%7B%202%20%7D%3D%5Cfrac%20%7B%2025%20%7D%7B%2016%20%7D%20%5C%5C%20%5C%5C%20%5Csqrt%20%7B%20%7B%20x%20%7D%5E%7B%202%20%7D%20%7D%20%3D%5Csqrt%20%7B%20%5Cfrac%20%7B%2025%20%7D%7B%2016%20%7D%20%20%7D%20%5C%5C%20%5C%5C%20x%3D%5Cquad%20%5Cpm%20%5Cfrac%20%7B%205%20%7D%7B%204%20%7D%20)
![{ x }^{ 2 }=1\\ \\ \sqrt { { x }^{ 2 } } =\sqrt { 1 } \\ \\ x=\quad \pm 1](https://tex.z-dn.net/?f=%7B%20x%20%7D%5E%7B%202%20%7D%3D1%5C%5C%20%5C%5C%20%5Csqrt%20%7B%20%7B%20x%20%7D%5E%7B%202%20%7D%20%7D%20%3D%5Csqrt%20%7B%201%20%7D%20%5C%5C%20%5C%5C%20x%3D%5Cquad%20%5Cpm%201)
So the values x has are :
![\frac { 5 }{ 4 }](https://tex.z-dn.net/?f=%5Cfrac%20%7B%205%20%7D%7B%204%20%7D%20)
,
![-\frac { 5 }{ 4 }](https://tex.z-dn.net/?f=-%5Cfrac%20%7B%205%20%7D%7B%204%20%7D%20)
,
![1](https://tex.z-dn.net/?f=1)
and
![-1](https://tex.z-dn.net/?f=-1)
Solution set :
![x=\quad \left\{ \frac { 5 }{ 4 } \quad ,\quad -\frac { 5 }{ 4 } \quad ,\quad 1\quad ,\quad -1 \right\}](https://tex.z-dn.net/?f=x%3D%5Cquad%20%5Cleft%5C%7B%20%5Cfrac%20%7B%205%20%7D%7B%204%20%7D%20%5Cquad%20%2C%5Cquad%20-%5Cfrac%20%7B%205%20%7D%7B%204%20%7D%20%5Cquad%20%2C%5Cquad%201%5Cquad%20%2C%5Cquad%20-1%20%5Cright%5C%7D%20)
I hope this was clear enough. If not please ask :)
The boxes checked are right
a) true: In some instances, an irrational number can be added to an irrational number to become rational, but adding a rational number to one will not work
b) false: It only sometimes works
c) true: e.g. square roots --> (√5)² = 5
d) true: a rational number multiplied by any rational number will be rational
e) false: e.g. perfect squares --> √64 = 8
f) true: e.g. √(5/3) = (√5)/(√3) = (√5*√3)/3
The number of tickets to sell at the door is 48 tickets.
<u>Step-by-step explanation</u>:
- Cost of play at the door = $5
- Cost of play in advance = $4
- The theater club wants to raise at least $400 from the play.
- Number of tickets sold in advance = 40 tickets.
Let 'x' be the number of tickets need to sell at the door.
5x + 4(40) = 400
5x + 160 = 400
5x = 240
x = 240/5
x = 48 tickets
The number of tickets to sell at the door is 48 tickets.
The sales tax would be .79.
i don’t feel like explaining because i am lazy . have a happy day :)
Answer: 292
Step-by-step explanation:
Given
There are 240 members in a labor union
The number increases by 5% each year
After 1 year it is
![\Rightarrow 240+5\%\times 240\\\Rightarrow 240(1+0.05)](https://tex.z-dn.net/?f=%5CRightarrow%20240%2B5%5C%25%5Ctimes%20240%5C%5C%5CRightarrow%20240%281%2B0.05%29)
After 2 years
Similarly, after 4 years
![\Rightarrow 240(1+0.05)^4\\\Rightarrow 291.72\approx 292](https://tex.z-dn.net/?f=%5CRightarrow%20240%281%2B0.05%29%5E4%5C%5C%5CRightarrow%20291.72%5Capprox%20292)