IB Maths AI SLTopic 1 — Number & AlgebraPaper 2GDC essential~6 min read
Solving Equations using a GDC
Your GDC can solve almost any equation — quadratics, cubics, exponentials, even messy ones with no algebraic solution. Three tools to know: polynomial root finder, numerical solver, and graph intersection. Pick the right one.
📘 What you need to know
Polynomial Root Finder — for any polynomial = 0 (linear, quadratic, cubic, quartic…). Type in degree and coefficients; it gives ALL real roots at once.
Numerical Solver — for any equation. You type in the equation and an initial guess; it returns ONE root near your guess.
Graph Intersection — plot the two sides as y1 and y2, then use the “intersect” feature to find where they cross. Shows ALL solutions in the viewing window.
Always rearrange first if needed: get equation into the form the tool expects (=0 for root finder, or two sides for intersection).
Find ALL solutions: an equation may have multiple roots. Check the graph window covers them all.
Give answers to 3 s.f. unless the GDC shows an exact integer / fraction.
Three GDC tools — which to use
Equation type
Best tool
Why
Polynomial = 0 (degree 2, 3, 4…)
Polynomial Root Finder
Finds all roots in one go.
Exponential, logarithmic, mixed
Graph Intersection or Numerical Solver
Not polynomial — root finder won’t accept it.
Simultaneous linear equations
Linear System Solver
Direct — type in coefficients.
Anything weird with no algebraic form
Graph Intersection
Visual — easy to spot all solutions.
x² = 2ˣ has no neat algebraic solution — but the GDC finds three intersections easily. Always set a wide enough window to catch them all.
🧭 Recipe — solve any equation on the GDC
Rearrange if helpful: get it into f(x) = 0 form (for solver/root finder), or leave both sides for graph intersection.
Pick the tool: polynomial = 0 → Polynomial Root Finder. Anything else → Graph Intersection (safest, shows all roots).
Set a sensible window: x and y wide enough to see all crossings. Zoom out if unsure.
Use the GDC’s feature: “intersect”, “zero”, or “solve” — never just read off the graph by eye.
Round to 3 s.f. on the final answer, unless GDC shows exact.
Worked examples
WE 1
Quadratic — Polynomial Root Finder
Solve x² − 7x + 10 = 0.
It’s a polynomial = 0 → use Polynomial Root Finderdegree: 2coefficients: a=1, b=−7, c=10GDC returns both rootsx = 2 or x = 5exact integers — no rounding needed.
WE 2
Cubic — Polynomial Root Finder
Solve x³ − 6x² + 11x − 6 = 0.
Cubic = 0 → Polynomial Root Finderdegree: 3coefficients: 1, −6, 11, −6GDC returns three rootsx = 1, x = 2, x = 3root finder handles higher degrees just as easily — type all coefficients in order.
WE 3
Exponential — Graph Intersection
Solve 2ˣ = 50, giving your answer to 3 s.f.
Not a polynomial — graph the two sidesy₁ = 2^xy₂ = 50Use “intersect” — single crossing pointx = 5.6438…x ≈ 5.64 (3 s.f.)also solvable by hand: x = log₂(50). Both give the same number.
WE 4
No algebraic solution — Graph Intersection finds ALL
Solve x² = 2ˣ, giving each solution to 3 s.f.
Graph both sides; look for all crossingsy₁ = x², y₂ = 2^xWindow: x from −2 to 5 catches all 3 intersectionsuse “intersect” three times — left, middle, rightx ≈ −0.766, x = 2, x = 4if you only spot 2 intersections, your window is too narrow. ALWAYS sketch first to know how many to expect.
WE 5
Simultaneous linear — Linear System Solver
Solve simultaneously: 2x + 3y = 13 and x + y = 5.
GDC Linear System Solver2 unknowns, 2 equationsenter coefficients row by row: [2, 3 | 13] [1, 1 | 5]x = 2, y = 3faster than substitution. Same tool handles 3 unknowns / 3 equations in HL, AI HL.
WE 6
Real-world — when does a population exceed a target?
A population is modelled by P(t) = 500 × 1.05ᵗ. Find the smallest time t at which the population first exceeds 1000.
Set up500 × 1.05^t = 1000Graph intersectiony₁ = 500 × 1.05^xy₂ = 1000intersect → t = 14.2066…t ≈ 14.2 years (or 15 if whole years only)read the question — “first exceeds” with whole-year context would round UP to 15.
💡 Top tips
Polynomial = 0? Use the root finder — fastest, finds all roots.
Anything else? Default to graph intersection — visual, finds all solutions in the window.
Sketch first: estimate how many solutions to expect, then check your GDC finds them all.
Show your method in working: “y₁ = …, y₂ = …, intersect” earns method marks even if your final number is slightly off.
⚠ Common mistakes
Missing solutions: window too narrow, missed the negative root or the high-x root. Always zoom out.
Reading the graph by eye: “looks like x = 3” is not acceptable — use the GDC’s “intersect” or “zero” function for an exact reading.
Wrong tool: trying to use Polynomial Root Finder on 2ˣ = 50 — it only accepts polynomials.
Numerical Solver returns just one root: if there are multiple, change the initial guess to find the others, OR switch to graph intersection.
That wraps the Number Toolkit. Up next: Exponentials & Logs — now that you can SOLVE exponential equations on the GDC, the next note shows how to solve them BY HAND using logarithm rules.
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