PanMaths.comFree Printable Math Formula Reference Sheet
Quadratic Formula
Algebra • Grades 8, 9, 10, 11
Algebra
Quadratic Formula
The go-to method for solving any quadratic equation.
The Formula
x = (-b ± √b²-4ac) / 2a
Used for: Solving quadratic polynomial equations of the standard form ax² + bx + c = 0
aQuadratic coefficient (a ≠ 0)
bLinear coefficient
cConstant term
ΔDiscriminant: b² - 4ac
⚡Live Interactive Quick Solver
Instant CalculationRoots of 1x² + -5x + 6 = 0:x = 3.00, x = 2.00
Discriminant (Δ = b² - 4ac): 1 (2 real roots)
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
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🎯 The Discriminant: b² - 4ac
The expression under the square root tells you what kind of solutions to expect:
✓✓
b² - 4ac > 0Two distinct real solutions
✓
b² - 4ac = 0One repeated real solution
∅
b² - 4ac < 0Two complex solutions
📝 Worked Examples
Two Real Solutions
Solve:
x² - 5x + 6 = 01
Identify coefficients
a = 1, b = -5, c = 62
Calculate discriminant
b² - 4ac = 25 - 24 = 13
Apply formula
x = (5 ± √1) / 24
Find solutions
x = 3 or x = 2Answer: x = 3 and x = 2
One Real Solution
Solve:
x² - 6x + 9 = 01
Identify
a = 1, b = -6, c = 92
Discriminant
b² - 4ac = 36 - 36 = 03
Apply formula
x = 6/2 = 3Answer: x = 3 (repeated root)
Complex Solutions
Solve:
x² + 2x + 5 = 01
Identify
a = 1, b = 2, c = 52
Discriminant
b² - 4ac = 4 - 20 = -163
Apply formula
x = (-2 ± 4i) / 2Answer: x = -1 + 2i and x = -1 - 2i
💡 Pro Tips
Check your signs
The formula has -b, so if b is negative, it becomes positive!
Discriminant first
Calculate b² - 4ac before the full formula to know what to expect
Simplify radicals
Always simplify √(b² - 4ac) before dividing by 2a
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