Using Calculators to Solve Equations (Cambridge (CIE) IGCSE International Maths)

Revision Note

Author

Mark Curtis

Expertise

Maths

Using Calculators to Solve Equations

How do I find points of intersection using my graphic display calculator?

You can use your graphic display calculator to draw two graphs on the same axes, $y = f_{1} (x)$ and $y = f_{2} (x)$ , then find the coordinates of the points of intersection
- These are the points where the two graphs meet

First draw the graph of $y = f_{1} (x)$ on your graphic display calculator
- Read the revision note on Using Calculators to Sketch Graphs to see how
Then add a second graph, $y = f_{2} (x)$ , on to the same axes of the graph above
- You may find the tab button is a shortcut for adding a second graph

Now navigate to "analyze graph" and select "intersection"
- On some models, press G-Solv and then INTSECT instead
- This finds the coordinates of the points of intersection
  - You may have to click before and after the intersection to help the calculator search for it
  - You may have to use the left and right buttons to select which intersection point you are finding

How do I solve equations using my graphic display calculator?

You can solve equations of the form $f_{1} (x) = f_{2} (x)$ using a graphic display calculator

$f_{1} (x)$ is the left-hand side of the equation, $f_{2} (x)$ is the right-hand side

The solutions to the equation $f_{1} (x) = f_{2} (x)$ are the x-coordinates of the points of intersection of the two graphs $y = f_{1} (x)$ and $y = f_{2} (x)$
- For example, to solve $x^{2} + 3 x + 1 = 2 x + 1$ using a graphic display calculator
  - draw $y = x^{2} + 3 x + 1$ on your calculator and add on $y = 2 x + 1$
  - Find the two points of intersection, which are (-1, -1) and (0, 1)
  - The solutions to $x^{2} + 3 x + 1 = 2 x + 1$ are the x-coordinates of these points
  - so $x = - 1$ or $x = 0$ are the solutions

Points of intersection between a curve and a line

What situations should I be familiar with?

Whilst you may be asked to solve any equation of the form $f_{1} (x) = f_{2} (x)$ , there are a few familiar situations that you should recognise
Solving a function equal to zero: $f_{1} (x) = 0$
- For example, to solve $x^{2} + 3 x + 1 = 0$
  - Using above, the solutions are the x-coordinates of the points of intersection of $y = x^{2} + 3 x + 1$ and $y = 0$ (since $f_{2} (x) = 0$ )
  - You need to recognise $y = 0$ as the x-axis, so the solutions are the x-intercepts of $y = x^{2} + 3 x + 1$
Solving a function equal to a constant (number): $f_{1} (x) = k$
- For example, to solve $x^{2} + 3 x + 1 = 1$
  - The solutions are the x-coordinates of the points of intersection of $y = x^{2} + 3 x + 1$ and $y = 1$ (since $f_{2} (x) = 1$ )
  - You need to recognise $y = 1$ as a horizontal line that cuts the y-axis at 1
To solve a different equation like $x^{2} + 3 x - 4 = 0$ using the original graph $y = x^{2} + 3 x + 1$ you need to rearrange the different equation to get "graph = ..."
- Add / subtract terms to both sides
  - For example, add 5 to both sides of $x^{2} + 3 x - 4 = 0$
  - $x^{2} + 3 x - 4 + 5 = 0 + 5$ giving $x^{2} + 3 x + 1 = 5$ which can now be done as above

Is this method the same as using the equation solver on my graphic display calculator?

The method above for solving equations of the form $f_{1} (x) = f_{2} (x)$ using the x-coordinate of intersection is called a graphical method
However, the equation $f_{1} (x) = f_{2} (x)$ can be rearranged by bringing all terms to one side
- $f_{1} (x) - f_{2} (x) = 0$ (or $f_{2} (x) - f_{1} (x) = 0$ )
  - E.g. $x^{2} + 3 x + 1 = 2 x + 1$ becomes $x^{2} + x = 0$
- If this forms a quadratic or cubic equation, you can use the equation solver on your calculator
  - This is no longer a graphical method but now an algebraic method
  - Look for the option for solving a polynomial
- Select the degree (or order) of the equation you are solving
  - When solving a quadratic, the degree (or order) is 2
  - When solving a cubic, the degree (or order) is 3
Type in the coefficients of the equation you are solving
- This is usually in the format $a_{2} x^{2} + a_{1} x + a_{0} = 0$ (or $a x^{2} + b x + c = 0$ )
- Then press enter (or solve) and view the solutions
  - This will give the same solutions as those from the graphical method

Exam Tip

If asked to solve an equation using a graphical method, you must do it by finding points of intersections between two graphs (not by using the equation solver).

Worked Example

Draw the graph of $y = x^{3} + 2 x^{2} + 1$ on your calculator.

Use a graphical method to solve the equation $x^{3} + 2 x^{3} + 1 = x + 2$ , giving your answers correct to 1 decimal place.

Start a new graph on your calculator and enter $f_{1} (x) = x^{3} + 2 x^{2} + 1$ to generate the graph
Add on the second graph of $f_{2} (x) = x + 2$

Select the option to "analyze" the graph (this may be labelled as G-Solv)
Choose the option "intersections" to find the coordinates of the 3 points of intersection

The solutions to the equation are the x-coordinates of the points of intersection

Drawing a cubic graph and a straight line graph on a graphic display calculator to find the points of intersection

$x = - 2.2$ , $x = - 0.6$ or $x = 0.8$ to 1 d.p.

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