Key Idea: When equations are too complex to solve by hand, you use your GDC to find the answer graphically. The key skill is setting up the problem correctly, then reading and interpreting what the graph shows. Two methods: find where two functions intersect, or find where one function crosses zero (its roots).
Three things IB tests on this topic:
📐 Two methods — same idea
Both methods are equivalent — you can always rearrange and solve either way. Solving f(x) = g(x) is the same as finding roots of h(x) = f(x) − g(x). Choose whichever the GDC makes easier. On most GDCs, finding intersections between two entered functions is the most direct approach.
✏️ Worked examples
Simultaneous equations (one non-linear)
Solve: y = x² − 2x and y = x + 4
Step by step:
Enter Y₁ = x² − 2x and Y₂ = x + 4 on GDC
Adjust window to see both curves (try x: −3 to 5, y: −5 to 15)
Find intersection: x = −1 (y = 3) and x = 4 (y = 8)
Final answer:
x = −1 or x = 4
Find roots of a cubic
Find all roots of f(x) = x³ − 4x + 1
Step by step:
Enter Y₁ = x³ − 4x + 1 on GDC
Graph shows 3 x-intercepts — zoom to see each one
Find each zero: x ≈ −2.11, x ≈ 0.25, x ≈ 1.86 (3 s.f.)
Final answer:
x ≈ −2.11, 0.25, 1.86
Context — break-even point
Two cost functions: C₁ = 50 + 2x, C₂ = 100 + x. Find where they cost the same.
Step by step:
Set equal: 50 + 2x = 100 + x → rearrange: x = 50 (can also use GDC)
Enter Y₁ = 50 + 2x and Y₂ = 100 + x, find intersection
Intersection at x = 50 — both cost $150 at that point
Final answer:
Equal cost when x = 50
Sketch the graph — even a rough sketch in your working earns a method mark and shows you know how many solutions to expect. Window settings matter — if you can't see the crossing or root, zoom out or manually set axes. Try x: −5 to 5 first, then adjust. How many roots? Quadratic: up to 2. Cubic: up to 3. The graph tells you immediately. Paper 1: You may be asked to read roots from a given graph, or set up (but not solve) an equation from context. You will not solve non-linear equations by hand.