Iteration for GCSE and A Level |

**Aim:** To use iteration to find an approximate solution to the equation x² - x - 5 = 0.

**Step 1:** Rearrange the equation to get the iterative form, x² = x + 5..

**Step 2:** Divide both sides of the equation by x to get x = 1 + 5/x - the iterative form.

**Step 3:** Now write it in this form: x_{n+1} = 1 + 5/x_{n}

**Step 4:** Use an initial value to trigger off iteration such as x_{0} = 1.

**Step 5:** We can now use the iterative formula. We can start with an initial guess of x_{0}=1 and then use the iterative formula to repeat, until we get a value that becomes relatively constant.

**Step 6:** Approximate the value to the required decimal places and it is a root of the quadratic equation.

x(0) = 1

x_{1}= 6.0

x

_{2}= 1.83333333333x

_{3}= 3.72727272727
x

_{4}= 2.34146341463x

_{5}= 3.13541666667x

_{6}= 2.59468438538
x

_{7}= 2.92701664533
x

_{8}= 2.708223972
x

_{9}= 2.84622839606
x

_{10}= 2.75671074286x

_{11}= 2.81375576416x

_{12}= 2.77698436505
x

_{13}= 2.80051427834
x

_{14}) = 2.78538636231
x

_{15}= 2.79508310504
x

_{16}= 2.78885557677
x

_{17}= 2.79285010011
x

_{18}= 2.79028584449
x

_{19}= 2.79193110623
x

_{20}= 2.79087513616As you can see, the solution/root is getting closer to 2.79. Therefore, the answer is x = 2.79 to 2 d.p.

You can practise the above interactively with the following Python programme:

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