Solving equations: algebraic and graphical methods

IB Mathematics: Applications and Interpretation HL· 2.9 Solving equations by algebraic and graphical methods· 10 min read

1. Algebraic methods for polynomial and rational equations★★☆☆☆⏱ 8 min

📘 Definition

Solution of an equation

For $f (x) = 0$

A value $x = a$ that makes the equation true, i.e. $f (a) = 0$ when the equation is rearranged to equal zero

Example:

$x = 2$ is a solution to $2 x - 4 = 0$ , since $2 (2) - 4 = 0$

Algebraic methods produce exact roots for many common equation types. For linear equations $a x + b = 0$ , the solution is always $x = - \frac{b}{a}$ . For quadratic equations $a x^{2} + b x + c = 0$ , we can use factorization, completing the square, or the quadratic formula to find exact roots.

📐 Worked Example

Solve $\frac{2 x}{x - 1} - \frac{3}{x + 1} = 2$ , find all valid solutions

1
Multiply both sides by the common denominator $(x - 1) (x + 1)$ to eliminate fractions:
2
$2x(x+1) - 3(x-1) = 2(x-1)(x+1)$
3
Expand all terms:
4
$2x^2 + 2x - 3x + 3 = 2x^2 - 2$
5
Simplify by collecting like terms:
6
$-x + 5 = 0 \implies x = 5$
7
Check for extraneous solutions by substituting back into the original equation:
8
$\text{LHS} = \frac{2(5)}{5-1} - \frac{3}{5+1} = 2.5 - 0.5 = 2 = \text{RHS}$
9
$x = 5$ is a valid solution

Exam tip:

Always check for extraneous solutions when you multiply by an expression containing the unknown, this is required for full marks on IB exams.

2. Algebraic methods for exponential and logarithmic equations★★★☆☆⏱ 10 min

Exponential equations with the unknown in the exponent can often be solved exactly using logarithm properties. We use the power rule $ln (a^{b}) = b ln (a)$ to bring the exponent down to a linear term, then solve for the unknown.

📐 Worked Example

Solve $3^{2 x - 1} = 7^{x + 2}$ for $x$ , give your answer to 3 decimal places

1
Take the natural logarithm of both sides:
2
$\ln\left(3^{2x -1}\right) = \ln\left(7^{x+2}\right)$
3
Apply the power rule for logarithms:
4
$(2x -1)\ln 3 = (x + 2)\ln 7$
5
Expand and collect terms with $x$ on the left-hand side:
6
$2x \ln 3 - \ln 3 = x \ln 7 + 2\ln 7 \\ x(2\ln 3 - \ln 7) = 2\ln 7 + \ln 3$
7
Solve for $x$ by dividing both sides:
8
$x = \frac{2\ln 7 + \ln 3}{2\ln 3 - \ln 7}$
9
Calculate using a calculator to get the final result:
10
$x \approx 7.052$

✓ Quick check

What is the correct first step to solve $2 e^{3 x} = 10$ ?

What is the correct first step?
- Divide both sides by 2 then take natural log
- Take natural log then divide by 2
- Set $3 x = ln (10)$
- Divide 10 by $2 e$

3. Graphical methods for approximate root finding★★☆☆☆⏱ 7 min

Many non-linear equations (such as $e^{- x} = x$ or $x^{3} - 2 x - 5 = 0$ ) cannot be solved exactly with algebraic methods. For these, we use graphical methods with your GDC to find accurate approximate roots.

There are two common approaches: 1) Rearrange the equation to $f (x) = 0$ , then roots are the $x$ -intercepts of $y = f (x)$ . 2) Split the equation into $f (x) = g (x)$ , then roots are the $x$ -coordinates of the intersection of $y = f (x)$ and $y = g (x)$ .

📐 Worked Example

Find an approximate solution to $e^{- x} = x$ using the intersection method

1
Split the equation into two functions to graph: $y_{1} = e^{- x}$ and $y_{2} = x$
2
Plot both functions on your GDC, they intersect at exactly one point for $x > 0$
3
Use the GDC's 'intersect' tool to find the coordinates of the crossing point
4
Read the $x$ -coordinate of the intersection to get the solution: $x \approx 0.567$
5
Verify: $e^{- 0.567} \approx 0.567$ , which matches the right-hand side

Exam tip:

Practice using the 'zero' and 'intersect' tools on your GDC before the exam, this saves significant time and avoids errors.

4. Comparing algebraic and graphical methods★★★☆☆⏱ 6 min

Methods compared

Choose your method based on the form of the equation and the required output:

Algebraic

Exact method for standard solvable equation types

+ Pros: Produces exact, analytic roots; No calculator required for basic cases

− Cons: Only works for specific equation types; Can produce extraneous solutions

Graphical

Approximate method for any graphed equation

+ Pros: Works for any equation you can plot; Shows the number of roots visually

− Cons: Only gives approximate roots; Requires a graphic display calculator

5. Common Pitfalls

Wrong move:

Forgetting to check for extraneous solutions after solving rational or logarithmic equations

Why:

Multiplying both sides by an expression containing $x$ can introduce values that make the original denominator zero or log argument negative

Correct move:

Always substitute all obtained solutions back into the original equation to confirm they satisfy it

Wrong move:

Missing multiple roots when only using an algebraic method

Why:

Higher-degree polynomials and non-linear equations often have more roots than the initial algebraic manipulation finds

Correct move:

Plot the function on your GDC to confirm how many roots exist before you start solving

Wrong move:

Rounding intermediate steps when solving algebraically

Why:

Early rounding leads to inaccurate final answers, especially for exponential and logarithmic equations

Correct move:

Keep full calculator precision through all steps, only round the final answer to the required accuracy

Wrong move:

Reporting the $y$ -coordinate of an intersection as the solution

Why:

The solution to $f (x) = g (x)$ is the value of $x$ that makes the equation true

Correct move:

Always confirm you report the $x$ -coordinate of the intersection as your solution

6. Quick Reference Cheatsheet

Equation Type	Algebraic Method	Graphical Alternative
Linear $a x + b = 0$	$x = - b / a$	$x$ -intercept of $y = a x + b$
Quadratic $a x^{2} + b x + c = 0$	Factorize or quadratic formula	$x$ -intercepts of parabola
Rational with fractions	Multiply by denominator, check for extraneous roots	Intersection of LHS and RHS
Exponential $a^{k x + b} = c^{m x + n}$	Take log, solve linear for $x$	Intersection of two exponentials
Non-solvable non-linear	Not applicable	Find $x$ -intercept or intersection

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2025 · 1
Solve quadratic and exponential equations
2024 · 2
Graphical solution of cubic equation
2023 · 1
Identify extraneous solution of rational equation

What's Next

Solving equations is a foundational skill for all further topics in IB AI HL, from applied modelling to calculus and optimization. Mastering the choice between exact and approximate methods prepares you for real-world problem solving, where most applied equations cannot be solved exactly with algebraic techniques. This sub-topic builds directly on basic function knowledge and leads into core applied topics including systems of equations, numerical methods, and optimization of functions for real-world models.