Free tool · shows working
Every solver prints the full method, because method is what your paper actually marks. An answer with no working scores poorly even when it is correct.
Roots, discriminant and factored form of ax² + bx + c
ax² + bx + c = 0
Answer
x = 3 or x = 2
Short answer
How do you factorise a quadratic equation?
Check the discriminant first: b² − 4ac. If it is a perfect square, the quadratic factorises over the integers — find two numbers that multiply to a×c and add to b, split the middle term, and factor in pairs. If it is positive but not a perfect square, use the quadratic formula. If it is negative, there are no real roots at all, and you have saved yourself ten minutes of trying to find factors that do not exist.
Step by step
Quadratic factorisation, cubic solution by rational root and synthetic division, or inverse functions. Each prints the full working rather than only a result.
Coefficients are entered separately so the tool can show the substitution rather than guess at parsing. The example above each field shows the expected shape.
Most maths papers award the majority of their marks for method. An answer with no working scores poorly even when it is correct, so the steps are the part worth copying into your own notes.
For a quadratic, b² − 4ac tells you the nature of the roots before you solve: positive gives two real roots, zero one repeated root, negative none. For an inverse, confirm that f(f⁻¹(x)) = x.
The solver is a worked example, not a substitute. If you cannot reproduce the method unaided, you have the answer and not the technique — and the exam tests the technique.
Worked examples
These are produced by the same solver above, run when this page was built. Most maths papers award the majority of their marks for method, so the working is the part worth reading — the answer on its own is the least valuable line.
Answerx = -2 or x = -3
The discriminant is positive and a perfect square, which is exactly when a quadratic factorises over the integers.
Answerx = 3 or x = -1
A leading coefficient other than 1 does not change the method — it changes which pair of numbers you are looking for, because you split against a×c rather than c.
Answerx = -1 ± 2i (no real roots)
A negative discriminant means the parabola never reaches the x-axis. Check it before you spend time trying to factorise.
Answerx = 1; x = 3 or x = 2
The Rational Root Theorem finds the first root; synthetic division reduces what is left to a quadratic you already know how to solve.
Answerf⁻¹(x) = (x − 4) / 3
Verify an inverse by composing it: f(f⁻¹(x)) should return x exactly.
Find two numbers that multiply to a×c and add to b, split the middle term with them, then factor in pairs. If no integer pair exists, the quadratic does not factorise over the integers and you use the quadratic formula instead — which is exactly what the discriminant tells you before you waste time trying.
b² − 4ac determines the nature of the roots. Positive means two distinct real roots and the parabola crosses the x-axis twice. Zero means one repeated root and the parabola touches the axis. Negative means no real roots — the parabola never reaches the axis, and the solutions are complex.
Find one root first, usually by the Rational Root Theorem: test factors of the constant term divided by factors of the leading coefficient. Once you have a root r, divide by (x − r) using synthetic division, which leaves a quadratic you can solve normally.
Swap x and y, then solve for y. For f(x) = 3x + 4: write y = 3x + 4, swap to get x = 3y + 4, then rearrange to y = (x − 4)/3. Check your answer by confirming that f(f⁻¹(x)) = x.
Only if you learn the method, which is why every solver here prints the full working. Most maths papers award the majority of marks for method, so an answer with no working scores poorly even when it is right.
