How to solve equations, polynomials and simultaneous equations on the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II
On the fx-CG50: MENU → EQUATION … F3 (SOLVER).On the TI-84 Plus CE: apps → PlySmlt2 → 1: Poly Root Finder.On the TI-Nspire CX II: menu → Algebra → Numerical Solve.
Polynomial roots, simultaneous linear equations and any equation f(x) = g(x) with the numerical solver. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
Every AI paper and AA Paper 2 expect you to solve equations you cannot rearrange: a cubic in an optimisation question, 3 simultaneous equations from a quadratic model through three points, or ex = 3 − x. Write the equation in your working, then give the calculator's answer to 3 s.f.
Practise it: SL AI solving equations · SL AI systems of equations · HL AA systems of equations · HL AA polynomials.
Which IB course and paper
| Course | GDC papers | What you use it for |
|---|---|---|
| AA SL | Paper 2 (Paper 1 is calculator-free) | Equations you cannot rearrange (e.g. eˣ = 3 − x) and checking polynomial roots. |
| AA HL | Papers 2 and 3 (Paper 1 is calculator-free) | As SL, plus systems of up to 3 linear equations. |
| AI SL | Papers 1 and 2 | Polynomial equations and systems of up to 3 linear equations, solved with technology. |
| AI HL | Papers 1, 2 and 3 | As SL; systems can also be solved with matrices. |
A GDC is allowed on every AI paper and on AA Papers 2 and 3; AA Paper 1 is calculator-free, so learn the method by hand as well.
Topic notes: Systems of linear equations and Gaussian elimination (AI HL) · Systems of equations and partial fractions (AA HL) · Solving trigonometric equations in a given interval (AA SL) · Trigonometric equations (AA HL).
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Solving equations on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
Watch the key presses on the fx-CG5014 presses, one at a time
The walkthrough needs JavaScript. The same key presses are in the numbered steps below.
- MENU, then choose EQUATION (the icon labelled A: highlight it and press EXE).
- Polynomial: F2 (POLYNOMIAL), choose the degree (F2 = 3 for a cubic), type the coefficients 1, 0, −2, −5 pressing EXE after each (use (−) for negatives, and 0 for a missing term), then F1 (SOLVE).
- Simultaneous: EXIT back, F1 (SIMULTANEOUS), choose the number of unknowns (F2 = 3), type the coefficient matrix row by row (a, b, c, d for ax + by + cz = d), then F1 (SOLVE).
- Any other equation: F3 (SOLVER). Type e^x = 3 − X using SHIFT . for the = sign, press EXE, check the Lower and Upper search bounds, then F6 (SOLVE). Lft − Rgt should be 0 (or tiny).
- Complex roots of a polynomial only appear when SHIFT MENU (SET UP) → Complex Mode is a+bi; in Real mode you see just the real root(s).
- SOLVER finds one root, the one nearest your starting value; change X or the bounds to find another. For all roots, graph y = f(x) − g(x) and use G-Solve ROOT instead.
- The polynomial solver goes up to degree 6; simultaneous equations up to 6 unknowns.
Solving equations on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
Watch the key presses on the TI-84 Plus CE9 presses, one at a time
The walkthrough needs JavaScript. The same key presses are in the numbered steps below.
- Polynomial or simultaneous: apps → PlySmlt2 → 1: Poly Root Finder (or 2: Simult Eqn Solver). Set the degree (or the number of equations and unknowns) and press graph (NEXT).
- Type the coefficients: 1, 0, −2, −5 for x³ − 2x − 5 (use (−), not −, for negatives). Press graph (SOLVE).
- Any other equation: math, then ▲ to reach Solver… at the bottom of the MATH list, and enter. Type the equation as 0 = e^(X) − 3 + X (the Solver wants 0 = …), enter.
- Put a starting guess next to X =, then alpha enter (SOLVE). A small square marks the solved variable; left−rt should be 0.
- PlySmlt2 is preloaded on the CE. The Solver in the MATH menu always works.
- For complex roots in PlySmlt2 press mode inside the app and choose a+bi.
- IB exams: the IB says not to use Press-to-Test on the TI-84 Plus CE, because it blocks Apps such as PlySmlt2. You reset the RAM and Archive Vars instead, so PlySmlt2 is still there in the exam.
Solving equations on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS (the CX II CAS is a different model: check the exam rules before using one).
- Open a Calculator page: home/on → New Document (or ctrl doc → Add Calculator).
- Polynomial: menu → Algebra → Polynomial Tools → Find Roots of Polynomial. Enter the degree 3, OK, then type the coefficients (or the polynomial) and press enter: polyRoots(x^3−2x−5, x).
- Simultaneous: menu → Algebra → Solve System of Linear Equations. Enter 3 equations and the variables x, y, z, OK, type each equation into the template and press enter.
- Any other equation: menu → Algebra → Numerical Solve, then nSolve(e^(x)=3−x, x) and enter. Add a guess or bounds, e.g. nSolve(…, x, 0, 2), to choose which root.
- The non-CAS CX II has no symbolic solve(); nSolve gives one numerical root at a time.
- For complex polynomial roots use cPolyRoots( instead of polyRoots(.
For teachers: Show on the board Print a keystroke card Teaching GDC skills
Worked example
(a) Solve x³ − 2x − 5 = 0. (b) Solve 2x + 3y − z = 5, x − y + 2z = 3, 3x + y + z = 10. (c) Solve ex = 3 − x.
Answers (checked with Python)
(a) x = 2.09455 (the other two roots are complex: −1.04728 ± 1.13594i, shown only in complex mode). (b) x = 4.6, y = −2, z = −1.8. (c) x = 0.792060 (0.792 to 3 s.f.).
Now try it on an exam-style question
Use the keystrokes above on IB-style solving equations questions. Free to try, no sign-up.
Common calculator mistakes
- Missing terms: x³ − 2x − 5 has no x² term, so its coefficient 0 must be entered. Leaving it out solves a different equation.
- Using the subtraction key − instead of the negative key (−) gives a Syntax ERROR.
- Trig equations: solve in the right angle mode (radians unless the question uses degrees) and within the domain the question gives. The solver finds one root; the question may want all of them.
- Rounding too early: keep the full answer in memory (store it in A) if you use it again; only round the final answer to 3 s.f.
- Writing just the answer: show the equation you solved, e.g. "x³ − 2x − 5 = 0 ⇒ x = 2.09 (GDC)".
- A system with no unique solution: the solver reports an error or "infinite solutions". On AA HL that is usually the point of the question: say which case it is and why.
- Angle mode: calculus with trig functions (and most AA trig) needs radians; bearings and triangle questions use degrees. Check the mode before the first trig question of the paper.
- Accuracy: unless the question says otherwise, IB markschemes want answers exact or to 3 significant figures. Keep the full calculator value for later parts and round only the answer you write down (x = 2.094551… is written 2.09, not 2.1 or 2.0946).
If you get an error
fx-CG50: Ma ERROR
Ma ERROR Casio's Math ERROR: the result is outside the allowable calculation range, or you are trying to do an illegal mathematical operation, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check your input values and change them so the result is within the allowable range.
TI-84 Plus CE: ERR:DOMAIN, ERR:SYNTAX
ERR:DOMAIN A value given to a function or instruction is outside the values it accepts. In a regression, TI says it means the data do not suit the model you chose. Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Exponential and power models need positive data. Fix: Change the value that is out of range, or choose a model that suits the data.
ERR:SYNTAX The command contains a syntax error: a misplaced function, argument, bracket or comma. Check: TI points out that subtract − and negative (−) are different keys, and using the wrong one causes this error. Then check the brackets and commas. Fix: Find the misplaced function, argument, bracket or comma and correct it.
TI-Nspire CX II: Domain error, Syntax error
Domain error An argument must be in a specified domain, and this one is not (TI's example: rand(0)). Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Fix: Change the value that is out of range.
Syntax error The syntax checker found a problem. Check: A missing bracket or comma. Fix: Correct the expression and try again.
Do it in Desmos
Not in the exam. Desmos is not an approved calculator for IB exams, so never rely on it for Paper 2. It is excellent for learning, checking a GDC answer and building graphs for your Internal Assessment (IA).
- Type the left-hand side and the right-hand side as two separate graphs: y = e^x and y = 3 − x.
- Click the point where they cross: Desmos shows its coordinates (0.792, 2.208).
- For a polynomial, graph y = x³ − 2x − 5 and click where it crosses the x-axis.
- Simultaneous equations in two unknowns are two straight lines: click their intersection. Three unknowns need the Desmos Matrix Calculator (desmos.com/matrix).
Open desmos.com/calculator (free, in any browser) and type:
y=e^{x}y=3-xy=x^{3}-2x-5
Questions
Can I use a calculator for solving equations in the exam?
In IB Maths you can use one approved GDC on every AI paper and on AA Papers 2 and 3, in exam mode. AA Paper 1 is calculator-free, so learn the method by hand as well.
What are the fx-CG50 keystrokes for solving equations?
MENU, then choose EQUATION (the icon labelled A: highlight it and press EXE). Then Polynomial: F2 (POLYNOMIAL), choose the degree (F2 = 3 for a cubic), type the coefficients 1, 0, −2, −5 pressing EXE after each (use (−) for negatives, and 0 for a missing term), then F1 (SOLVE). Then Simultaneous: EXIT back, F1 (SIMULTANEOUS), choose the number of unknowns (F2 = 3), type the coefficient matrix row by row (a, b, c, d for ax + by + cz = d), then F1 (SOLVE). The full steps and the other calculators are in the keystrokes section above.
Does this work on other calculators?
Yes. This page has keystrokes for the Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II. The worked example has the same answers on every model, so use it to check.
What should I write as working?
Write the maths, not the key presses: say what you are finding in normal notation (the equation, the probability statement or the integral), then the calculator's answer exact or to 3 significant figures unless the question says otherwise. Keep full accuracy in later parts.
More calculator skills
Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the models shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. A calculator with no tab here has no checked steps for this skill yet. Worked-example answers were recomputed independently in Python.