Pietro MeloniPhD

IB Calculator Guide / Solving equations

GDC skill · nSolve · polynomial roots

Solving equations on your calculator: solver, polynomial roots and intersections

IB Maths AA and AI, SL and HL, every calculator paper. The single most used GDC skill after arithmetic.

On a calculator paper the IB does not want you to solve ex=3−xe^x = 3 - x by hand: it wants the answer to 3 s.f. with a sketch that shows how you found it. There are three tools for this, and knowing which to reach for saves minutes. The numeric solver takes any equation and one starting guess, and finds one solution. The polynomial tool finds all roots of a polynomial at once, including complex ones at HL. Graphing both sides and finding the intersection is slowest but shows how many solutions there are, which the other two do not.

The example used below

(a) Solve x3−4x2+x+3=0x^3 - 4x^2 + x + 3 = 0. (b) Solve ex=3−xe^{x} = 3 - x. (c) Find the number of solutions of sin⁡x=0.1x\sin x = 0.1x for x>0x > 0.

Answers

(a) x=−0.700,1.24,3.46x = -0.700, 1.24, 3.46. (b) x=0.792x = 0.792. (c) Three: x=2.85x = 2.85, 7.077.07 and 8.428.42. For x>10x > 10 the line 0.1x0.1x is above 1, so there are no more.

Choose your calculator

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TI-Nspire CX II

In one line. Calculator page: menu, 3 Algebra, 1 Numerical Solve for one equation; 3 Polynomial Tools, 1 Find Roots of Polynomial for all roots. Graphs page: menu, 6 Analyze Graph, 4 Intersection.

  1. 1

    All roots of a polynomial: in a Calculator page press menu, Algebra, Polynomial Tools, Find Roots of Polynomial. Set Degree 3, Roots Real (or Complex at HL), then type the coefficients 1, −4, 1, 3.

    menu3: Algebra3: Polynomial Tools1: Find Roots of PolynomialDegree: 3a3=1, a2=−4, a1=1, a0=3OK
  2. 2

    polyRoots(x³−4x²+x+3,x) = {−0.699628, 1.23912, 3.4605}. Write x=−0.700x = -0.700, 1.241.24, 3.463.46.

  3. 3

    One equation, one solution: menu, Algebra, Numerical Solve. Type the equation with the variable and a guess: nSolve(e^(x)=3−x, x, 1).

    menu3: Algebra1: Numerical SolvenSolve(e^(x)=3−x,x,1)enter
  4. 4

    Result 0.792059. Write x=0.792x = 0.792. To find a different solution change the guess, or add a range: nSolve(f(x)=g(x), x, 1) | x > 5.

  5. 5

    Counting solutions: add a Graphs page, type f1(x)=sin(x) and f2(x)=0.1x. Set the window (menu, 4 Window/Zoom, 1 Window Settings) to x from 0 to 12, y from −1.5 to 1.5.

    ctrldoc2: Add Graphsf1(x)=sin(x)f2(x)=0.1x
  6. 6

    menu, Analyze Graph, Intersection. Click a lower bound to the left of a crossing and an upper bound to the right. Repeat for each crossing: 2.85234, 7.06817 and 8.42320.

    menu6: Analyze Graph4: Intersectionlower boundupper bound

Typed form

  • polyRoots(x³−4x²+x+3, x)
  • cPolyRoots(...) for complex roots (HL)
  • nSolve(equation, x, guess)
  • zeros(x³−4x²+x+3, x) also works on the non-CAS model for polynomials

Where the TI-Nspire CX II trips people up

  • nSolve returns one solution only, the one nearest the guess. If the question says "solve", check with a graph that there are no others.
  • Make sure the angle mode is Radians (settings in the top bar, or ctrl, doc, 7 Settings) when the equation has trigonometric functions. The IB uses radians unless the question says degrees.
  • In Press-to-Test mode on a CAS model, solve( ) is disabled but nSolve( ) and polyRoots( ) still work.

Casio fx-CG50

In one line. MENU, Equation: F2 Polynomial for all roots, F3 Solver for any single equation. MENU, Graph, draw, then SHIFT F5 G-Solv, F5 INTSECT for crossings.

  1. 1

    All roots of a polynomial: open Equation, choose Polynomial, Degree 3, and type the coefficients 1, −4, 1, 3 (EXE after each). Press F1 SOLVE.

    MENUEquationF2 PolynomialF2 Degree 3a=1, b=−4, c=1, d=3F1 SOLVE
  2. 2

    X1 = 3.46050487, X2 = 1.239123278, X3 = −0.699628148. Write x=−0.700x = -0.700, 1.241.24, 3.463.46. Complex roots appear automatically if SET UP has Complex Mode a+bi.

  3. 3

    One equation: in Equation choose Solver. Type e^X = 3 − X (the = sign is SHIFT then the decimal point key), EXE, set a starting X, then F6 SOLV.

    MENUEquationF3 Solvere^X=3−XEXEX: 1F6 SOLV
  4. 4

    X = 0.7920592. Write x=0.792x = 0.792. Lft and Rgt show both sides of the equation at the solution; they should match.

  5. 5

    Counting solutions: open Graph, type Y1 = sin X and Y2 = 0.1X. Set the window with SHIFT F3 V-Window (Xmin 0, Xmax 12, Ymin −1.5, Ymax 1.5), then F6 DRAW.

    MENUGraphY1=sin XY2=0.1XSHIFT F3 V-WindowF6 DRAW
  6. 6

    SHIFT F5 opens G-Solv. F5 INTSECT jumps to the first intersection; press the right arrow for the next ones: X = 2.8523, 7.0682 and 8.4232.

    SHIFTF5 G-SolvF5 INTSECT▶ next

Typed form

  • G-Solv also has F1 ROOT (zeros of one curve), F2 MAX, F3 MIN, F4 Y-ICPT
  • Run-Matrix: SolveN(e^X=3−X) returns every solution in the window (OPTN, F4 CALC, F6, F2 SolveN)

Where the Casio fx-CG50 trips people up

  • If INTSECT says "Not Found", the crossing is outside the viewing window. Widen the window and draw again.
  • The Solver needs the equation typed with the = sign from SHIFT and the decimal point key, not a comparison from the menu.
  • Angle unit is in SHIFT MENU SET UP: Rad for IB unless the question says degrees.

TI-84 Plus CE

In one line. MATH, C Numeric Solver for one equation; APPS, PlySmlt2 for polynomial roots; Y=, GRAPH, then 2nd TRACE (CALC), 5 intersect for crossings.

  1. 1

    All roots of a polynomial: APPS, PlySmlt2, 1 Polynomial Root Finder. Set ORDER 3, REAL or a+bi, and press GRAPH (NEXT). Type the coefficients 1, −4, 1, 3 and press GRAPH (SOLVE).

    APPSPlySmlt21: Polynomial Root FinderORDER: 3NEXTa3=1, a2=−4, a1=1, a0=3SOLVE
  2. 2

    x1 = 3.4605049, x2 = 1.2391233, x3 = −.6996281. Write x=−0.700x = -0.700, 1.241.24, 3.463.46.

  3. 3

    One equation: MATH, C Numeric Solver (OS 5.x; it is 0 Solver on older OS). Type E1: e^(X) and E2: 3 − X, press GRAPH (OK), set X = 1, then ALPHA ENTER (SOLVE).

    MATHC: Numeric SolverE1: e^(X)E2: 3−XOKX=1ALPHAENTER (SOLVE)
  4. 4

    X = .79205920. Write x=0.792x = 0.792. The line "bound" lets you restrict the search when there are several solutions.

  5. 5

    Counting solutions: Y=, type Y1 = sin(X) and Y2 = 0.1X. WINDOW: Xmin 0, Xmax 12, Ymin −1.5, Ymax 1.5. GRAPH.

    Y=Y1=sin(X)Y2=0.1XWINDOWGRAPH
  6. 6

    2nd TRACE opens CALC. Choose 5 intersect. Press ENTER for First curve, ENTER for Second curve, move the cursor near the crossing and press ENTER for Guess. Repeat for the other crossings: X = 2.8523, 7.0682 and 8.4232.

    2ndTRACE (CALC)5: intersectENTERENTERmove cursorENTER

Typed form

  • CALC menu: 1 value, 2 zero, 3 minimum, 4 maximum, 5 intersect, 6 dy/dx, 7 ∫f(x)dx
  • Zeros of a single function: 2 zero with Left bound, Right bound, Guess

Where the TI-84 Plus CE trips people up

  • The Guess step matters: with several crossings the calculator returns the one nearest to the cursor when you press ENTER.
  • MODE must say RADIAN for trigonometric equations.
  • The Numeric Solver finds one solution. Use the graph to count them first.

Where the marks go, on any calculator

  • Giving one solution when there are two or three. Any equation with a trigonometric, exponential or polynomial side can have several; a quick sketch shows how many.
  • No sketch, no working. For a GDC solution the IB expects a sketch of the graphs (axes labelled, curves labelled, intersections marked) or the equation rearranged to f(x)=0f(x) = 0 with "using GDC" written. A bare answer with no evidence may score the final A mark only.
  • Solving in degrees. Check the angle mode before every trigonometric equation.
  • Rounding the solution and then substituting the rounded value in a later part. Store the full value (sto→ on TI, → on Casio) and reuse it.

What to write on the paper

  1. Write the equation you are solving and the method: "ex=3−xe^x = 3 - x, by GDC x=0.792x = 0.792". The sketch is the working.
  2. For a sketch: label the axes, label each curve with its equation, mark the intersection(s) with their coordinates.
  3. If the question says "solve exactly" or "by hand", the GDC is for checking only: a calculator answer scores nothing.
  4. Give all solutions in the required domain, and state the domain you used: "for 0<x<120 < x < 12, three solutions".

Questions students ask

Is a GDC answer with no working accepted?
Yes for a final A mark, often no for the method marks. The IB mark schemes for calculator papers usually say "(M1) for a sketch or for the equation set to zero, A1 for the answer". A sketch takes thirty seconds and protects the method mark.
What is the difference between nSolve, polyRoots and intersect?
nSolve (TI-Nspire), the Solver (Casio) and the Numeric Solver (TI-84) find one solution of any equation from a starting guess. polyRoots, Polynomial and PlySmlt2 find every root of a polynomial at once. Intersection on the graph screen works for any equation and shows how many solutions there are, but you have to repeat it for each crossing.
How do I solve a system of equations on the GDC?
TI-Nspire: menu, Algebra, Solve System of Linear Equations (for linear systems) or two graphs. Casio: Equation, F1 Simultaneous, choose the number of unknowns. TI-84: APPS, PlySmlt2, 2 Simultaneous Eqn Solver, or rref( on a matrix. For non-linear systems use the intersection of the two graphs.

Practise it

Other calculator skills

Written by Pietro Meloni, PhD, IB Mathematics and Physics tutor. Keystrokes checked on TI-Nspire CX II OS 6, Casio fx-CG50 OS 3 and TI-84 Plus CE OS 5; older systems may number the menus differently. Not affiliated with the International Baccalaureate Organization, Texas Instruments or Casio. Found a key that has moved? Write to pietro@pietromeloni.com.

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