In mathematics we learn methods to solve simple equations such as $2x+4=5$2x+4=5 and more difficult equations such as $x^2+6x+8=0$x2+6x+8=0. But how can we find the solution for more complex equations such as $x^42x=0$x4−2x=0
We could guess the solution for $x$x , substitute it into the equation then evaluate the equation to check if it is a true statement. If it is not true we will try another value for $x$x.
Solve $x^42x=0$x4−2x=0
Guess  $x$x  $=$=  $1$1  
Check does  $1^42$14−2  $=$=  $0$0  
$1$−1  $=$=  $0$0  not true 
Guess  $x$x  $=$=  $2$2 
Check does  $2^42\times2$24−2×2  $=$=  $0$0 
$164$16−4  $=$=  $0$0 

$12$12  $=$=  $0$0 
What is the problem with this process and what do we need to consider? What are suitable values to try for $x$x?
Another possible method could be to write $x$x as a function of $x$x ,$f$f $x$x . The equation can be rearranged as follows: $x=\frac{1}{2}x^4$x=12x4
In this case we can try a value for $x$x, evaluate a new value for$x$x and then check if our new value is close to the old value.
Pick an initial guess for $x$x and call this $x_0$x0 (are there any restrictions for values of $x$x?)
$x_1=\frac{1}{2}x_0^4$x1=12x40 , evaluate for $x_1$x1
Then use $x_1$x1as the next guess for $x$x
$x_2=\frac{1}{2}x_1^4$x2=12x41 , evaluate for $x_2$x2
We can use an excel spreadsheet to perform these iteration calculations
The spreadsheet table could look like this
$x_i$xi  $x_i+1$xi+1 
1  * 
**  
$x_i+1$xi+1  $x_1$x1 
1  0.5 
0.5  0.03125 
0.03125  0.0000004768371582 
$x_i+1$xi+1  $x_i$xi 
10  5000 
5000  312500000000000 
312500000000000  4.76837E+57 
4.76837E+57  2.58494E+230 
Are the values of x converging (approaching a particular number) or diverging (not approaching a number  getting further from each successive value for $x$x)? Will the loop in our flowchart /program ever terminate? This method of solving the equation has not been successful.
The original equation can be rearranged as follows: $\sqrt[4]{2x_i}$^{4}√2xi
Change your flowchart for this new formula
Set up a spreadsheet in excel using this new formula and use the process of iteration to calculate successive values of $x$x. Fill in the cells below.
x  $\sqrt[4]{2x_i}$^{4}√2xi 
1  
x  $\sqrt[4]{2x_i}$^{4}√2xi 
1  1.189207115 
1.189207115  1.241857812 
1.241857812  1.255380757 
1.255380757  1.25878444 
1.25878444  1.259636801 
1.259636801  1.259849982 
1.259849982  1.259903282 
1.259903282  1.259916608 
1.259916608  1.259919939 
The solution for $x$x correct to $4$4 decimal places is $1.2599$1.2599
$x$x  $\sqrt[4]{2x_i}$^{4}√2xi 
10  2.114742527 
2.114742527  1.434075029 
1.434075029  1.30136903 
1.30136903  1.270157613 
1.270157613  1.26247243 
1.26247243  1.260558411 
1.260558411  1.26008036 
1.26008036  1.259960876 
1.259960876  1.259931006 
1.259931006  1.259923539 
1.259923539  1.259921672 
The solution for $x$x correct to $4$4 decimal places is $1.2599$1.2599