Using Calculators to Solve Equations (Cambridge International IGCSE 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, $y=straight f_ 1 (x)$ and $y=straight f_ 2 (x)$, then find the coordinates of the points of intersection

  • These are the points where the two graphs meet

• First draw the graph of $y=straight f_ 1 (x)$ on your graphic display calculator

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

• Then add a second graph, $y=straight f_ 2 (x)$, 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 $\text{straight} f_ 1 (x)=straight f_ 2 (x)$ using a graphic display calculator

• straight f_ 1 (x) is the left-hand side of the equation, straight f_ 2 (x) is the right-hand side

• The solutions to the equation $\text{straight} f_ 1 (x)=straight f_ 2 (x)$ are the x-coordinates of the points of intersection of the two graphs $y=straight f_ 1 (x)$ and $y=straight f_ 2 (x)$

  • For example, to solve $x^{2}+3x+1=2x+1$ using a graphic display calculator

    • draw $y=x^{2}+3x+1$ on your calculator and add on $y=2x+1$

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

    • The solutions to $x^{2}+3x+1=2x+1$ are the x-coordinates of these points

    • so $x=-1$ or $x=0$ are the solutions

What situations should I be familiar with?

• Whilst you may be asked to solve any equation of the form $\text{straight} f_ 1 (x)=straight f_ 2 (x)$, there are a few familiar situations that you should recognise

• Solving a function equal to zero: straight f_ 1 (x)=0

  • For example, to solve $x^{2}+3x+1=0$

    • Using above, the solutions are the x-coordinates of the points of intersection of $y=x^{2}+3x+1$ and $y=0$ (since straight f_ 2 (x)=0)
    • You need to recognise $y=0$ as the x-axis , so the solutions are the x-intercepts of $y=x^{2}+3x+1$

• Solving a function equal to a constant (number): straight f_ 1 (x)=k

  • For example, to solve $x^{2}+3x+1=1$

    • The solutions are the x-coordinates of the points of intersection of $y=x^{2}+3x+1$ and $y=1$ (since straight 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 $x^{2}+3x-4=0$ using the original graph $y=x^{2}+3x+1$ you need to rearrange the different equation to get "graph = ..."

  • Add / subtract terms to both sides

    • For example, add 5 to both sides of $x^{2}+3x-4=0$
    • $x^{2}+3x-4+5=0+5$ giving $x^{2}+3x+1=5$ 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 $\text{straight} f_ 1 (x)=straight f_ 2 (x)$ using the x-coordinate of intersection is called a graphical method

• However, the equation $\text{straight} f_ 1 (x)=straight f_ 2 (x)$ can be rearranged by bringing all terms to one side

  • $\text{straight} f_ 1 (x)−straight f_ 2 (x)=0$ (or $\text{straight} f_ 2 (x)−straight f_ 1 (x)=0$)

    • E.g. $x^{2}+3x+1=2x+1$ 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 $ax^{2}+bx+c=0$)

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

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

Exam Hint 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 $y=x^{3}+2x^{2}+1$ on your calculator.

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

Start a new graph on your calculator and enter straight f_ 1 (x)=x³+2x²+1 to generate the graph Add on the second graph of straight 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

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