Problem solving with iteration | Maths | Quizalize & Oak National Academy

Starter quiz

What are the solutions to the quadratic equation $x^2-3x-10=0$ ?

$x=-3$ and $x=-10$

$x=3$ and $x=-10$

$x=-2$ and $x=-5$

$x=2$ and $x=-5$

$x=-2$ and $x=5$ ✓
What are the approximate solutions to the quadratic equation $x^2+2x-5=0$ ?

$x=1.22$ and $x=-2.68$ to 3 significant figures

$x=-1.22$ and $x=2.68$ to 3 significant figures

$x=1.45$ and $x=-3.45$ to 3 significant figures ✓

$x=-1.45$ and $x=3.45$ to 3 significant figures
Which value does the iterative formula $x_{n+1}=\frac{8}{x_n-2}$ converge on to 1 significant figure when $x_0=5$ ?

-3

-2 ✓

-1

0.8

12
Using the iterative formula $x_{n+1}=\frac{(x_n)^3+1}{2}$ and $x_0=0$ , which of these is a solution to $x^3-2x+1=0$ correct to 2 significant figures?

$x=0.56$

$x=0.59$

$x=0.62$ ✓

$x=0.67$
Between which sets of values is there definitely a solution to $x^2-5x-1=0$ ?

Between $x=-2$ and $x=-1$

Between $x=-1$ and $x=0$ ✓

Between $x=4.8$ and $x=4.9$

Between $x=5.1$ and $x=5.2$ ✓
Which of these are solutions to $x^2+2x-6=0$ correct to the stated degree of accuracy?

$x=2$ to 1 significant figure ✓

$x=1.5$ to 2 significant figure

$x=-3$ to 1 significant figure

$x=-4$ to 1 significant figure ✓

Exit quiz

Laura has used the equation $x^2+x-12=0$ to write the iterative formula $x_{n+1}=12-(x_n)^2$ . If 3 is an exact solution, what should happen when she uses $x_0=3$ ?

An error will occur.

The first iteration will equal 0.

The first iteration will equal 3. ✓

The outputs of the formula will eventually converge on 3.

The iterative formula will not converge.
Which of these could be iterative formulae to find a solution to the equation $x^2-5x+4=0$ ?

$x_{n+1} = \sqrt{4-5x_n}$

$x_{n+1} = -\sqrt{5x_n+4}$

$x_{n+1}=\frac{(x_n)^2}{5} +4$

$x_{n+1}=-\frac{4}{x_n-5}$ ✓
Which of these could be iterative formulae to find a solution to the equation $x^2+2x-6=0$ ?

$x_{n+1}=\frac{6}{x_n+2}$ ✓

$x_{n+1}=3-\frac{(x_n)^2}{2}$ ✓

$x_{n+1} = \sqrt{2x_n-6}$

$x_{n+1} = (x_n)^2+2(x_n)-6$
Using the iterative formula $x_{n+1}=\frac{6}{x_n+2}$ which values of $x_0$ converge on a solution to the equation $x^2+2x-6=0$ ?

$x_0=-5$

$x_0=-3$ ✓

$x_0=-2$

$x_0=0$ ✓

$x_0=2$ ✓
Which of these iterative formulae converge on the approximate solution $x=0.459$ to the equation $x^2-7x+3=0$ when $x_0=2$ ?

$x_{n+1}=-\frac{3}{x_n-7}$ ✓

$x_{n+1}=\sqrt{7x_n-3}$

$x_{n+1}=-\sqrt{7x_n-3}$

$x_{n+1}=\frac{(x_n)^2+3}{7}$ ✓
$Using the graph (or otherwise) which values of <Math>x_0</Math> will converge on a solution to the equation <Math>x^2-7x+3=0</Math> using the iterative formula <Math>x_{n+1}=\frac{(x_n)^2+3}{7}</Math>?$

Using the graph (or otherwise) which values of $x_0$ will converge on a solution to the equation $x^2-7x+3=0$ using the iterative formula $x_{n+1}=\frac{(x_n)^2+3}{7}$ ?

$x_0=-7$

$x_0=-5$ ✓

$x_0=0$ ✓

$x_0=5$ ✓

$x_0=8$

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Lesson Details

Key learning points

Quadratics can have their solutions estimated using iteration
This can be helpful to quickly determine where the solutions are likely to be
It can be used to check any solutions

Common misconception

An iterative formula can always be used to find the solutions to an equation.

This lesson addresses limitations of iteration. Some formulas will cause an error becasue we are unable to evaluate the square root of a negative or divide by 0. Other formulas may not converge. Different values of $x_0$ produce different results.

Keywords

Iteration - Iteration is the repeated application of a function or process in which the output of each iteration is used as the input for the next iteration.