Signs of a solution | Maths | Quizalize & Oak National Academy

Starter quiz

A value has been rounded to 2.8 to 2 significant figures. What is the lower bound of this value?

2

2.5

2.7

2.75 ✓
A value has been rounded to 2.8 to 2 significant figures. What is the upper bound of this value?

'2.85' ✓
What is the value of $x^3-2x+5$ when $x=2$ ?

'9' ✓
What is the value of $x^3-2x+5$ when $x=-2$ ?

'1' ✓
Match the letter representing each graph to the equation of the graph.

A

⇔

$y=x^2$ ✓

B

⇔

$y=x^3$ ✓

C

⇔

$y=-x^2$ ✓

D

⇔

$y=\frac{1}{x}$ ✓
Using the iterative formula $x_{n+1}=\sqrt[3]{4(x_n)^2-1}$ and $x_0=5$ there is a solution to $x^3-4x^2+1=0$ correct to 2 significant figures when $x=$ ______ .

'3.9' ✓

Exit quiz

Using the table of values shown for the equation $y=x^3-4x+2$ which of these statements are definitely true?

There is a solution to the equation $x^3-4x+2=0$ between $x=0$ and $x=0.5$

There is a solution to the equation $x^3-4x+2=0$ between $x=0.5$ and $x=1$ ✓

There is a solution to the equation $x^3-4x+2=0$ between $x=1$ and $x=2$ ✓

There is a solution to the equation $x^3-4x+2=0$ between $x=1$ and $x=1.2$

There is a solution to the equation $x^3-4x+2=0$ between $x=1.6$ and $x=2$ ✓
Between which sets of values is there definitely a solution to $x^3-5x-1=0$ ?

Between $x=-3$ and $x=-2$ ✓

Between $x=-2.3$ and $x=-2.2$

Between $x=-0.4$ and $x=-0.5$

Between $x=2.3$ and $x=2.4$ ✓
When might a change in sign not indicate there is a solution between two values?

When the equation being solved is not equal to zero. ✓

When one input is negative but the other is positive.

When the values are really far apart.

When the graph has an asymptote and doesn't cross the axis between those points. ✓
The image shows Jacob's workings to test his approximate solution to the equation $x^3-2x=0$ . What do you think his solution was?

$x=1.3$

$x=1.35$

$x=1.355$

$x=1.4$ ✓

$x=1.45$
To see if $x=-1.4$ is a solution to the equation $x^3-2x=0$ to 2 significant figures which values should we test to see if there is a change in sign?

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

$x=-1.3$ and $x=-1.5$

$x=-1.35$ and $x=-1.45$ ✓

$x=-1.4$ and $x=-1.45$
Which of these are solutions to $-x^2+3x-1=0$ correct to the stated degree of accuracy?

$x=2$ to 1 significant figure

$x= 3$ to 1 significant figure ✓

$x=0.38$ to 2 significant figures ✓

$x=2.5$ to 1 decimal place

Worksheet

Loading worksheet ...

Presentation

Loading presentation ...

Video

Lesson Details

Key learning points

If the value is a solution, then substituting it into the equation should mean the equation evaluates to 0
Since the solution is an approximation, it is unlikely to be exactly 0
By substituting the upper and lower bonds of the estimated value into the equation, you can tell if the estimate is good
If the two values produce one positive and one negative value, this implies the graph would cross the x axis
The x intercept would be a solution to the equation

Common misconception

A change in sign always indicates a solution. No change in sign means no solution.

A change in sign only indicates a solution if the equation is equal to zero and if the graph of the equation crosses the $x$ axis between those points. Where there is a repeated root or two solutions between them there may be no change in sign.

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.
Lower bound - The lower bound for a rounded number is the smallest value that the number could have taken prior to being rounded.
Upper bound - The upper bound for a rounded number is the smallest value that would round up to the next rounded value.