How to solve equations, polynomials and simultaneous equations on the Casio fx-CG50
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.
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).
- 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.
- 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. Depending on the settings chosen when Press-to-Test is turned on, some apps can be disabled: check yours before the exam (the Solver in the MATH menu always works).
- For complex roots in PlySmlt2 press mode inside the app and choose a+bi.
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 not allowed in IB, A Level or IGCSE exams.
- 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(.
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.).
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)".
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).
The interactive graph loads here when you scroll to it.
Or open desmos.com/calculator and type:
y=e^{x}y=3-xy=x^{3}-2x-5
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 OS versions 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. Worked-example answers were recomputed independently in Python.