Use substitution to solve transformed equations
Recognise a repeated expression, substitute a temporary variable, solve the quadratic and reverse the substitution within its domain.
Section 4 · Equations, inequalities and graphs
Cambridge IGCSE Additional Mathematics 0606, 2028–2030 syllabus, Version 1. Paper 1 is non-calculator; Paper 2 requires a scientific calculator. These original lessons provide self-checked topic practice, not a Cambridge grade or a timed paper. Full-paper assessment and human specialist review are still outstanding.
Before you begin
- Cambridge assumes knowledge of IGCSE Mathematics or an equivalent course. Check the domain before using an inverse or logarithm, and retain exact values where the question asks for them.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which substitution is natural in x⁴ − 5x² + 4 = 0?
What you will learn
- Choose a substitution that turns an equation into a quadratic
- Solve the resulting quadratic
- Reject substituted roots that violate the substitution's range
Choose a repeated inner expression
Equations in x⁴ and x² can become quadratic by setting u = x². Equations with repeated ln x, eˣ or another expression can use the same strategy.
Solve the quadratic in u, then reverse the substitution. The temporary variable may have restrictions: x² ≥ 0, eˣ > 0 and ln x requires x > 0.
Check candidates in the original domain
For x⁴ − 5x² + 4 = 0, set u = x². Then u² − 5u + 4 = 0 gives u = 1 or 4; both are allowed because u ≥ 0.
For logarithmic substitutions, a quadratic root that makes ln x negative is not automatically invalid, but the resulting x must still be positive. Always test the original equation and domain.
Pause and explain
After solving u = x², which real u-values can correspond to x?
Worked example
Solve x⁴ − 5x² + 4 = 0 for real x.
Show the worked solution
- Set u = x², so u² − 5u + 4 = 0.
- Factor: (u − 1)(u − 4) = 0, giving u = 1 or 4.
- Return to x² = 1 or x² = 4 and take both real square roots.
Answer x = −2, −1, 1 or 2.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Solve x⁴ − 10x² + 9 = 0.
Give me a hint
Let u = x² and remember u cannot be negative.
Compare my reasoning
- Form u² − 10u + 9 = 0.
- Factor: (u − 1)(u − 9) = 0.
- Return to x² = 1 or x² = 9.
x = −3, −1, 1 or 3.
Look for these in your work
- Substitution correct
- Both u-values valid
- Both signs of x included
Solve (ln x)² − 3 ln x + 2 = 0 for x > 0.
Give me a hint
Use u = ln x, then exponentiate each solution.
Compare my reasoning
- Set u = ln x: u² − 3u + 2 = 0.
- Factor to get u = 1 or 2.
- Thus x = e or e², both positive.
x = e or e².
Look for these in your work
- Quadratic solved
- Logarithm reversed correctly
- Domain checked
A population model produces 3^(2t) − 5·3^t + 4 = 0. Find its real time solutions t.
Give me a hint
Set u = 3^t and remember this quantity is positive.
Compare my reasoning
- Let u = 3^t > 0, giving u² − 5u + 4 = 0.
- Factor to get u = 1 or 4; both are positive.
- Therefore 3^t = 1 or 3^t = 4.
t = 0 or t = log₃4.
Look for these in your work
- Repeated exponential identified
- Positive roots retained
- Logarithmic solution stated
Common mistakes
- Stopping after solving for the temporary variable
- Taking only positive square roots
- Keeping substituted values that cannot occur in the original domain
What can you explain now?
If u = eˣ, what values can u take?
Compare with the explanation
u > 0
An exponential is strictly positive, so zero and negative substituted roots are impossible.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Describe substitution, the temporary variable's range and the final check in the original equation.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.