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
Use it to check factorised roots, solve simultaneous equations in mechanics and statistics, and find the root you then locate by the sign-change method in numerical methods. Exam answers need the working too: the calculator only confirms.
Practise it: Algebra and functions · Numerical methods.
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(.
Solving equations on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
- MENU A (Equation/Func).
- Polynomial: 2 (Polynomial), then the degree (3 for a cubic). Type 1 = 0 = −2 = −5 =, then press = repeatedly to step through x₁, x₂, x₃ (and the turning points).
- Simultaneous: MENU A 1 (Simul Equation), choose 3 unknowns, fill in the coefficient table with = after each number, then = to show x, y, z.
- Any other equation: MENU 1 (Calculate). Type e^x = 3 − x (the = is ALPHA CALC), then SHIFT CALC (SOLVE), give a start value and press =.
- Polynomials up to degree 4 and up to 4 simultaneous unknowns. For quadratics it also shows the vertex (x-Value Minimum/Maximum) after the roots.
- Complex roots appear only if Complex Roots is set to a+bi in SHIFT MENU (SETUP).
Solving equations on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
- HOME, choose the Equation app with the arrow keys, OK.
- Polynomial: choose Polynomial, pick the degree, type the coefficients with EXE after each, then EXE to step through the roots.
- Simultaneous: choose Simul Equation, pick the number of unknowns, fill the coefficient table, EXE to show the solution.
- Any other equation: in the Equation app choose Solver, type the equation using the = from VARIABLE or the on-screen template, EXE, give a starting value and EXE again.
- Polynomials up to degree 4 and up to 4 unknowns. Where Solver sits in the menu can look slightly different from the fx-991EX: the fx-991CW User's Guide (Equation app chapter) is the reference.
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 allowed in A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- 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.
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- 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.