GCSE 9-1 Iterations » More

Solving equations



Solving equations by iteration

Iteration involves rearranging the equation you are trying to solve to give an iteration formula. This is then used repeatedly (using an estimate to start with) to get closer and closer to the answer.

Example:
Find the iteration formula for   x2 – 3x + 1 = 0
Rearrange the equation:    3x = x2 + 1
Divide by 3: x =  (x2 + 1)/3
Relabel x's:  xn+1 = (xn2 + 1)/3
This means that x1 is found using x0 i.e. n=0

Solve x2 = 3x – 1 = 0 using the iterative formula.
Set your calculator, to use the previous answer in the next iteration.

Use x0 = 1 to start in the equation: x1 = (x02 + 1) / 3
i.e. x1 = (12 + 1) / 3 = 0.6666
Find x2 with x1 = 0.6666
i.e. x2 = (0.66662 + 1) / 3 = 0.481
Repeat to find xn+1 using xn until you get to 2 dp.
x = 0.38 (to 2 dp)


But this is only one of the roots for x2 – 3x + 1 = 0

The other root can be found with a different iterative formula.
Q1: Find another iterative formula for x2 – 3x + 1 = 0
(hint: rearrange differently)

Q2:Use the formula to find the other root to 2 dp

ANS

Solve cubic equation with with 3 roots

The equation x3 – 10x + 6 = 0 has 2 positive and one negative root.

a) Show that x3 – 10x + 6 = 0 can be written as xn+1 = 3√(10xn – 6) .

b) Taking x0 = 0.5 use the iterative formula xn+1 = 3√(10xn – 6) to find x1, x2 and x3 to 2 dp

c) Find the negative root of x3 – 10x + 6 = 0 to 2dp

d) Find one of the positive roots of x3 – 10x + 6 = 0 by taking x0 = 1 to 2 dp

ANS

d) Find another iterative formula for x3 – 10x + 6 = 0.

e) Use this formula to find the other positive root to 2dp

ANS

Iterative formula

Q1: Show that x = 1 + 14/(x – 5) is a rearrangement of the equation
x2 – 6x – 9 = 0

Q1b: Use the iterative formula xn+1 = 1 + 14/(xn – 5) with a starting value of x1 = 1 to find a root of the equation x2 – 6x – 9 correct to 2dp

ANS

More solving equations

Q1: Show that x3 + 5x = 4 has a solution between 0 and 1

Q1b: Show that x3 + 5x = 4 can be arranged to give: x = 4/5 – x3/5

Q1c: Use the iterative formula xn+1 = 4/5 – xn3/5 twice to find as estimate for the solution of
x3 + 5x = 4.   Start with x0= 0.5

ANS

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