When an equation is too messy to solve by hand — like ex = x2 — your GDC takes over. Plot the right graphs, find where they meet, and read off the answer.
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What you need to know
To solve f(x) = g(x) graphically, find where the two curves intersect
Or rearrange to f(x) − g(x) = 0 and find the roots of the difference
Use this when analytical methods fail or aren’t required
The number of solutions = the number of intersection points
GDC answers are approximate — for an exact answer, you must solve analytically
Allowed on Paper 2 (calculator paper) unless the question demands an algebraic method
The Two Methods
To solve f(x) = g(x) graphically, there are two equivalent approaches — choose whichever is easier to plot:
1
Plot Both Curves
Plot y = f(x) and y = g(x). The x-coordinates of the intersection points are the solutions.
2
Plot the Difference
Plot y = f(x) − g(x). The x-intercepts (roots) are the solutions.
Both methods give the same answer. Method 1 is more visual; Method 2 keeps everything on a single curve. For example, the solutions to x3 = 5ex are the same as the roots of y = x3 − 5ex.
When to Use Graphical Methods
Some equations can’t be solved analytically — there’s no algebraic trick. That’s when you reach for the GDC:
Analytical or Graphical?
Mixed function types (e.g. ex = x2)
→
Graphical
Polynomial of degree 5 or higher
→
Graphical
Question asks for an exact value
→
Analytical
Question asks for the number of solutions
→
Graphical (just count)
Standard quadratic, log, or exponential equation
→
Either works
Paper 1 (no calculator) or “algebraic” instruction
→
Analytical
Using Your GDC
Enter the function(s) in graphing mode (y1, y2, …).
Plot the graph and adjust the window so all features are visible.
Use the “intersect” function to find intersection points (Method 1) or the “root/zero” function for x-intercepts (Method 2).
Read off the x-coordinate — that’s your solution.
Round appropriately — usually 3 significant figures unless told otherwise.
Watch the window! If you can’t see all the intersections, zoom out or change the axes. The default window can hide solutions that are far from the origin.
The ex = x2 Example
Here are the two graphs side-by-side. Notice how they intersect at exactly one point (around x ≈ −0.703):
Method 1: y = ex and y = x2
Or using Method 2 — plot the difference and find the root:
Method 2: y = ex − x2
Worked Examples
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Example 1 — Sketch and find the solution (full SME problem)
Two functions are defined by f(x) = ex and g(x) = x2.
(a) Sketch the graph of y = ex − x2.
Use GDC to identify key features.x-intercept: x ≈ −0.7034y-intercept: f(0) − g(0) = 1 − 0 = 1, so (0, 1)Sketch a curve that crosses the x-axis at ≈ (−0.703, 0), passes through (0, 1), and rises sharply for x > 0.Label both intercepts on your sketch.
(b) Hence find the solution to ex = x2.
Step 1: rearrange to ex − x² = 0.Step 2: the solution is the x-intercept of y = ex − x² (the graph from part a).x ≈ −0.703 (3 s.f.)“Hence” means use part (a) — don’t start over.
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Example 2 — Number of solutions
For the equation x3 − x = 2, write down the number of real solutions.
Answer:
Step 1: plot y = x³ − x and y = 2 on your GDC.Step 2: count the number of intersection points.y = 2 is a horizontal line. y = x³ − x is a cubic.They intersect at exactly 1 point (around x ≈ 1.52).1 real solutionNumber of solutions = number of intersection points. No need to find the actual values.
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Example 3 — Find intersection coordinates
Two functions are f(x) = x3 − x and g(x) = 4x. Find the coordinates of the points where y = f(x) and y = g(x) intersect.
Answer:
Step 1: plot both graphs on your GDC.Step 2: use the intersect function.First intersection: x ≈ −1.60, y ≈ −2.50Second intersection: x ≈ 1.60, y ≈ 2.50(−1.60, −2.50) and (1.60, 2.50)Always check for multiple intersections — adjust the window if needed.
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Example 4 — Solve using intersection coordinates
Using your answer to Example 3, write down the solutions to x3 − x = 4x.
Answer:
The solutions are the x-coordinates of the intersection points.x ≈ −1.60 and x ≈ 1.60Just the x-values — the y-coordinates aren’t part of the solution to a single-variable equation.
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Example 5 — Polynomial that can’t be solved analytically
Find the real solutions to x5 − x + 1 = 0 to 3 significant figures.
Answer:
A polynomial of degree 5 — no analytical formula exists. Use the GDC.Step 1: plot y = x⁵ − x + 1.Step 2: find the x-intercepts (roots).Only one root visible: x ≈ −1.17x ≈ −1.17 (3 s.f.)There’s only one real root — the other four solutions are complex (not part of AA SL).
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Tips
Adjust the window until you can see all intersections. The default zoom often misses far-off solutions.
Use Method 2 when a single curve is easier to read than two crossing ones — particularly useful for counting roots.
For “number of solutions”, just count intersection points — you don’t need to find them.
Always sketch the graph as part of your working, even if you used the GDC. It shows the examiner your reasoning.
Round to 3 s.f. by default unless the question specifies otherwise.
If the question says “hence”, use the previous part’s graph rather than starting from scratch.
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Common mistakes
Reporting y-coordinates as solutions. Solutions to f(x) = g(x) are just the x-values, not the (x, y) pairs.
Missing intersections because the GDC window is too small. Always zoom out before committing to “no solutions” or “one solution.”
Using GDC when the question says “exact value.” An approximate decimal isn’t exact — switch to an analytical method.
Reading too few decimal places. “1.6” usually isn’t accurate enough — get 3 significant figures.
Confusing roots with intersections. A “root” is where a single curve hits the x-axis (Method 2). An “intersection” is where two curves meet (Method 1). Both give the same answer to f(x) = g(x).
Using graphical method on Paper 1. Paper 1 is no-calculator — graphical solving isn’t an option.
Final word: Graphical methods turn impossible-looking equations into a quick GDC plot. Choose Method 1 or 2 by what’s easier to plot, label the key points, and report the x-coordinate (rounded sensibly) as your solution.
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