Solve quadratics by inspection, factorisation or formula
Use inspection for a simple square equation · Apply the zero-product property · Use the quadratic formula and interpret the discriminant
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Factorisation
- Square roots and surds
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Use inspection for a simple square equation
- Apply the zero-product property
- Use the quadratic formula and interpret the discriminant
Choose an efficient method
For x² = 25, inspection gives x = ±5. For a product equal to zero, at least one factor is zero: (x−2)(x+3) = 0 gives 2 or −3. Put a quadratic into ax²+bx+c = 0 before applying these methods.
Formula with signs
For a ≠ 0, x = [−b ± √(b²−4ac)]/(2a). Use the signed values of a, b and c. A positive discriminant gives two distinct real roots, zero gives one repeated root, and a negative discriminant gives no real roots. The 0444 quadratic formula was supplied; you still need to identify coefficients and substitute accurately.
Keep exact radical forms unless a decimal is requested. Check roots in the original equation; a context may then select only a feasible root.
Sources for this lesson
Pause and explain
What does a negative discriminant mean over the real numbers?
Worked example
Solve x²−4x+1 = 0.
Show the worked solution
- a = 1, b = −4, c = 1; discriminant = 16−4 = 12.
- x = (4 ± √12)/2 = 2 ± √3.
Answer x = 2 ± √3.
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 x² = 25 over the real numbers.
Give me a hint
Both signs square to a positive value.
Compare my reasoning
- 5² = 25 and (−5)² = 25.
- There are exactly those two real roots.
x = ±5.
Look for these in your work
- Include both roots
- Check by squaring
Solve 2x²+x−3 = 0 by factorisation.
Give me a hint
Look for a product yielding the middle coefficient 1.
Compare my reasoning
- 2x²+x−3 = (2x+3)(x−1).
- Zero-product property gives −3/2 and 1.
x = −3/2 or 1.
Look for these in your work
- Verify the factorisation
- Set each factor to zero
Solve 3x²−2x−2 = 0 using the formula.
Give me a hint
Keep b = −2 and c = −2 signed.
Compare my reasoning
- Discriminant = 4+24 = 28.
- x = (2 ± √28)/6 = (1 ± √7)/3.
x = (1 ± √7)/3.
Look for these in your work
- Use the full denominator
- Simplify the radical exactly
Common mistakes
- Use −b, not b, in the numerator.
- A factor is set to zero only when the product equals zero.
What can you explain now?
What does a negative discriminant mean over the real numbers?
Compare with the explanation
No real roots.
The formula would require a real square root of a negative value.
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Make it stick
What does a negative discriminant mean over the real numbers?
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.