Using Calculators to Solve Equations (Cambridge (CIE) IGCSE International 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, and , then find the coordinates of the points of intersection

  • These are the points where the two graphs meet

• First draw the graph of on your graphic display calculator

  • Read the revision note on Using Calculators to Sketch Graphs to see how

• Then add a second graph, , 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 using a graphic display calculator

• is the left-hand side of the equation, is the right-hand side

• The solutions to the equation are the x-coordinates of the points of intersection of the two graphs and

  • For example, to solve using a graphic display calculator

    • draw on your calculator and add on

    • Find the two points of intersection, which are (-1, -1) and (0, 1)

    • The solutions to are the x-coordinates of these points

    • so or are the solutions

What situations should I be familiar with?

• Whilst you may be asked to solve any equation of the form , there are a few familiar situations that you should recognise

• Solving a function equal to zero:

  • For example, to solve

    • Using above, the solutions are the x-coordinates of the points of intersection of and (since )

    • You need to recognise as the x-axis, so the solutions are the x-intercepts of

• Solving a function equal to a constant (number):

  • For example, to solve

    • The solutions are the x-coordinates of the points of intersection of and (since )

    • You need to recognise as a horizontal line that cuts the y-axis at 1

• To solve a different equation like using the original graph you need to rearrange the different equation to get "graph = ..."

  • Add / subtract terms to both sides

    • For example, add 5 to both sides of

    • giving 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 using the x-coordinate of intersection is called a graphical method

• However, the equation can be rearranged by bringing all terms to one side

  • (or )

    • E.g. becomes

  • 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 (or )

  • Then press enter (or solve) and view the solutions

    • This will give the same solutions as those from the graphical method