Solve an equation numerically (including multiple solutions)
Faster and safer than algebra for messy equations - and essential in AI, where many equations can't be solved by hand. The trick is getting all solutions, not just one.
In short: Solving an equation numerically means using the GDC's solver or graph tools to find roots as decimal values when algebra is impractical. You enter the equation, or graph both sides and find the intersection, and the calculator returns approximate solutions. Always check that you have found every solution in the interval the question allows.
When you'd use this
The equation can't be rearranged into a form you can solve by hand, e.g. it mixes x and eˣ.
A question explicitly says "solve numerically" or "using your GDC".
You need every solution in a given interval, not just the first one you find.
Checking an algebraic solution is correct, or that you haven't missed a root.
At a glance: TI-84 Plus CE, Casio fx-CG50 and TI-Nspire CX compared
TI-84 Plus CE
Casio fx-CG50 and fx-CG100
TI-Nspire CX
Key sequence
MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.
Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.
Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.
On a TI-84 Plus CE
Graph f(x) first so you can see how many solutions exist and roughly where they are.
Rearrange so everything is on one side: f(x) = 0 - or graph both sides as separate functions and find intersections.
MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution.
For transcendental equations (e.g. e^x = 3x), graph both sides, count crossings, then use the intersection tool for each one.
Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
Tip: For equations like sin x = 0.5 over an interval, use the graph + intersection method rather than the equation solver - it's faster and less likely to miss roots.
On a Casio fx-CG50 and fx-CG100
Graph f(x) first so you can see how many solutions exist and roughly where they are.
Rearrange so everything is on one side: f(x) = 0 - or graph both sides as separate functions and find intersections.
Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach.
For transcendental equations (e.g. e^x = 3x), graph both sides, count crossings, then use the intersection tool for each one.
Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
Tip: For equations like sin x = 0.5 over an interval, use the graph + intersection method rather than the equation solver - it's faster and less likely to miss roots.
On a TI-Nspire CX
Graph f(x) first so you can see how many solutions exist and roughly where they are.
Rearrange so everything is on one side: f(x) = 0 - or graph both sides as separate functions and find intersections.
Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root.
For transcendental equations (e.g. e^x = 3x), graph both sides, count crossings, then use the intersection tool for each one.
Always verify each solution by substituting back into the original equation.
Tip: The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect.
Tip: For equations like sin x = 0.5 over an interval, use the graph + intersection method rather than the equation solver - it's faster and less likely to miss roots.
Here's a real IB-style question that uses exactly this technique.
MediumCalculatorPaper 2[3 marks]
Solve eˣ = 3x for 0 < x < 3, giving each solution to 3 significant figures.
x = 0.619 and x = 1.51
Mark it
Correct3 / 3 marks
Worked solution & mark scheme:
M1 Graph y = eˣ and y = 3x, or rearrange to eˣ − 3x = 0
A1 x = 0.619
A1 x = 1.51
Common questions
When would I need to solve an equation numerically, including multiple solutions in IB Maths?
Faster and safer than algebra for messy equations - and essential in AI, where many equations can't be solved by hand. The trick is getting all solutions, not just one. The equation can't be rearranged into a form you can solve by hand, e.g. it mixes x and eˣ. A question explicitly says "solve numerically" or "using your GDC".
How do I solve an equation numerically, including multiple solutions on a TI-84 Plus CE?
1. Graph f(x) first so you can see how many solutions exist and roughly where they are. 2. Rearrange so everything is on one side: f(x) = 0 - or graph both sides as separate functions and find intersections. 3. MATH → Solver: enter the expression, type a starting guess close to one root, press ALPHA + ENTER. Move the guess to near a different root and repeat for each solution. 4. For transcendental equations (e.g. e^x = 3x), graph both sides, count crossings, then use the intersection tool for each one. 5. Always verify each solution by substituting back into the original equation.
How do I solve an equation numerically, including multiple solutions on a Casio fx-CG50 and fx-CG100?
1. Graph f(x) first so you can see how many solutions exist and roughly where they are. 2. Rearrange so everything is on one side: f(x) = 0 - or graph both sides as separate functions and find intersections. 3. Run-Matrix → SolveN(f(x), x) returns all real roots at once; or use the Equation app for a visual approach. 4. For transcendental equations (e.g. e^x = 3x), graph both sides, count crossings, then use the intersection tool for each one. 5. Always verify each solution by substituting back into the original equation.
How do I solve an equation numerically, including multiple solutions on a TI-Nspire CX?
1. Graph f(x) first so you can see how many solutions exist and roughly where they are. 2. Rearrange so everything is on one side: f(x) = 0 - or graph both sides as separate functions and find intersections. 3. Type nSolve(f(x)=0, x, guess) - include a guess or interval e.g. nSolve(f(x)=0, x, 2) or nSolve(f(x)=0, x, {1,5}) to target a specific root. 4. For transcendental equations (e.g. e^x = 3x), graph both sides, count crossings, then use the intersection tool for each one. 5. Always verify each solution by substituting back into the original equation.
What should I watch out for when I solve an equation numerically, including multiple solutions?
The solver finds ONE root near your starting guess - change the guess to find others. The graph shows you how many to expect. For equations like sin x = 0.5 over an interval, use the graph + intersection method rather than the equation solver - it's faster and less likely to miss roots.
How are marks awarded when I solve an equation numerically, including multiple solutions in an IB exam?
In the worked example on this page (3 marks, Paper 2), the marks are: M1: Graph y = eˣ and y = 3x, or rearrange to eˣ − 3x = 0; A1: x = 0.619; A1: x = 1.51.
