# Solving equations graphically and analytically with a GDC

> IB Mathematics Analysis and Approaches SL · IB AA SL
> Source: https://www.owlsprep.com/study/ib-math-aa-sl-u2-solving-equations-graphically-and-analytically/

This guide covers paired analytical and GDC graphical methods to solve linear, quadratic, rational and simple exponential equations for IB AA SL Paper 1 and Paper 2.

**Prerequisites:** [Basic algebraic rearrangement and expansion of polynomials](https://www.owlsprep.com/study/ib-math-aa-sl-u2-algebraic-manipulation-polynomials/); [Understanding of function graphs and intercepts](https://www.owlsprep.com/study/ib-math-aa-sl-u2-function-graph-features/)

## Learning objectives

- Identify when to use analytical vs GDC graphical methods for equation solving
- Execute step-by-step algebraic solving for common function types
- Use GDC features to find roots, intersections and approximate solutions efficiently
- Verify solutions across both methods to avoid careless errors

## 1. Distinguishing Analytical vs GDC Graphical Methods

Before solving any equation in an IB exam, first confirm if the question permits a GDC, and whether an exact or approximate solution is requested. Paper 1 questions will almost always require full analytical working for exact answers, while Paper 2 questions can often be solved much faster using GDC graphical features.

**Exact analytical solution** — A solution expressed as an integer, fraction or simplified radical that requires no rounding, with zero approximation error

*Example:* $x=3$ or $x=\frac{1+\sqrt{5}}{2}$

**Worked example:** State which solving method is most appropriate for the following exam question: 'Solve $2x + 7 = 19$ for exact $x$'

1. Step 1: The question asks for an exact solution, no GDC permitted (Paper 1 context)
2. $$2x = 19 -7 = 12$$
3. $$x = 12 / 2 = 6$$

**Exam command terms**

IB exam command terms for equation solving have strict expectations:

- **Solve (exact)** — You must show full algebraic working, no GDC approximation allowed

- **Find the solution to 3 s.f.** — GDC graphical method is permitted, you only need to state the final 3 significant figure answer with minimal working

## 2. Step-by-Step Analytical Solving Workflow

For analytical solving, always rearrange your original equation to the form $f(x) = 0$ before factoring, quadratic formula, or other algebraic techniques. This reduces the chance of sign errors when moving terms across the equals sign.

**Derivation:** Derive the exact solution for $3x^2 - 5x - 2 = 0$

*Starting from:* Standard quadratic form $ax^2 + bx + c = 0$

1. Identify coefficients: $a=3$, $b=-5$, $c=-2$
2. $$Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 -4ac}}{2a}$$$
3. $$Substitute values: $x = \frac{5 \pm \sqrt{25 + 24}}{6} = \frac{5 \pm7}{6}$$$
4. Calculate both roots to get exact values

*Conclusion:* Exact solutions are $x=2$ and $x=-\frac{1}{3}$

**Worked example:** Analytically solve $\frac{2}{x} = x +1$ for exact $x$

1. Step 1: Note domain restriction $x \neq 0$
2. $$Multiply both sides by x: $2 = x^2 + x$$$
3. $$Rearrange to $x^2 +x -2 = 0$$$
4. $$Factor: $(x+2)(x-1) = 0$$$
5. Final exact solutions: $x=1$, $x=-2$ (both satisfy domain restriction)

## 3. GDC Graphical Solving Full Workflow

For GDC solving, you can use one of two standard workflows: either graph $y_1 = f(x)$ and $y_2 = 0$ then find the x-intercepts, or graph $y_1 = f(x)$ and $y_2 = g(x)$ then find their intersection points. Both methods are valid, but the intersection method is faster for equations that do not rearrange easily.

> **tip**
>
> Always adjust your GDC window to show all relevant roots before running the intersect tool, otherwise you may miss a valid solution that falls outside the default view range.

**Worked example:** Use a TI-84 GDC to solve $e^x = x +4$ to 3 significant figures

1. Step 1: Enter Y1 = e^x, Y2 = x +4 in the graph menu
2. Step 2: Adjust window to Xmin=-5, Xmax=5, Ymin=-2, Ymax=10 to show all intersections
3. Step 3: Run 2ND > CALC > intersect, select both curves, guess near the left intersection to get first root
4. Step 4: Repeat the intersect process for the right intersection to get second root
5. Final 3 s.f. solutions: $x \approx 1.75$, $x \approx -2.80$

**Check your understanding**

Test your GDC knowledge:

1. What is the most likely error if your GDC returns no intersection points?

   - Your equation has no real solutions
   - Your window range is too small
   - You entered the function incorrectly
   - All of the above

   *Why:* All three are common causes, always check your function input first then expand the window range before concluding no real solutions exist.

*Calculator:* graphing_only

## 4. Cross-Verifying Solutions Between Methods

IB exam markers award full marks only if your solution is consistent across working. If you solve a question analytically first, plug the solution back into your GDC graph to confirm it matches the root or intersection point, and vice versa.

**Comparing methods**

Compare the two solving methods for exam use:

- **Analytical Method** — Produces exact solutions, no rounding errors, required for Paper 1
  - Pros: Guarantees full marks for exact answer questions
  - Cons: Slow for complex functions, higher risk of algebraic errors

- **GDC Graphical Method** — Produces approximate 3 s.f. solutions in seconds, ideal for Paper 2
  - Pros: Very fast, eliminates algebraic manipulation errors
  - Cons: Cannot return exact radical or fraction solutions

## Common pitfalls

- **Wrong:** Using GDC approximation for a question that explicitly asks for an exact solution
  - Why it fails: IB markers will deduct all method marks even if your approximate value is close to the exact answer
  - Correct: Check the question for the word 'exact' first, use full algebraic working for these questions
- **Wrong:** Forgetting to exclude invalid solutions that violate domain restrictions after solving a rational equation
  - Why it fails: You may include a root that makes the original denominator zero, which is not a valid real solution
  - Correct: After finding all candidate solutions, substitute each back into the original equation to confirm it is defined
- **Wrong:** Rounding your GDC solution to 2 significant figures instead of the required 3
  - Why it fails: IB AA SL has a strict 3 s.f. default for all approximate answers unless stated otherwise
  - Correct: Double check your final GDC output and round to 3 significant figures before writing it down
- **Wrong:** Not showing at least one line of working even when you solve the full question on GDC
  - Why it fails: You will lose method marks if you only write the final answer with no indication you used the GDC intersect tool
  - Correct: Write one short line e.g. 'Using GDC intersection function, x = 2.34' to document your process

## Cheatsheet

| Method | Best for | Required Output | Exam Paper |
| --- | --- | --- | --- |
| Analytical solving | Exact solutions, Paper 1 | Simplified exact value | Paper 1 (No GDC) |
| GDC root finding | Find f(x)=0 roots | 3 s.f. approximate value | Paper 2 (GDC) |
| GDC intersection | Solve f(x)=g(x) | 3 s.f. approximate value | Paper 2 (GDC) |
| Cross verification | Check all solutions | Confirm no missing roots | Both papers |

## What's next

Now that you can reliably solve equations using both analytical and GDC graphical methods, you will build on this skill to solve systems of equations with multiple variables, a common 6-8 mark question on IB AA SL Paper 2. You will also learn how to identify and solve for the number of roots of a function using discriminant analysis, which lets you confirm if your GDC has found every possible real solution before you submit your final answer. These skills together will cut down your exam solving time significantly and reduce careless error rates for high mark function questions.

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