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Roots, discriminant and factored form of ax² + bx + c
ax² + bx + c = 0
Working
- Equation: x² - 5x + 6 = 0, so a = 1, b = -5, c = 6
- Discriminant: b² − 4ac = (-5)² − 4(1)(6) = 1
- √1 = 1
- x = (−b ± √(b²−4ac)) / 2a = (5 ± 1) / 2
- x₁ = 3, x₂ = 2
- Factored form: (x − 3)(x − 2)
- Both roots are integers, so this factorises cleanly — worth checking by expanding back out.
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.
Frequently asked questions
- How do you factorise a quadratic?
- 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.
- What does the discriminant tell you?
- 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.
- How do you solve a cubic equation?
- 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.
- How do you find an inverse function?
- 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.
- Will showing the answer help me in an exam?
- 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.