What is the formula for Quadratic Formula?

The mathematical formula for Quadratic Formula is x = (-b ± √(b²-4ac)) / 2a. It is used for solving quadratic polynomial equations of the standard form ax² + bx + c = 0.

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Quadratic Formula

Algebra • Grades 8, 9, 10, 11

Algebra

Quadratic Formula

Q: What do the variables in the Quadratic Formula formula represent?

In the formula x = (-b ± √(b²-4ac)) / 2a: a = Quadratic coefficient (a ≠ 0), b = Linear coefficient, c = Constant term, Δ = Discriminant: b² - 4ac.

Q: How do you calculate Quadratic Formula step-by-step?

To calculate quadratic formula: 1. Identify the given values (a = Quadratic coefficient (a ≠ 0), b = Linear coefficient, c = Constant term, Δ = Discriminant: b² - 4ac). 2. Substitute the values into the formula x = (-b ± √(b²-4ac)) / 2a. 3. Simplify and evaluate step-by-step.

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 Calculation

Roots 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 > 0

Two distinct real solutions

✓b² - 4ac = 0

One repeated real solution

∅b² - 4ac < 0

Two complex solutions

📝 Worked Examples

Two Real Solutions

Solve: x² - 5x + 6 = 0

Identify coefficientsa = 1, b = -5, c = 6

Calculate discriminantb² - 4ac = 25 - 24 = 1

Apply formulax = (5 ± √1) / 2

Find solutionsx = 3 or x = 2

Answer: x = 3 and x = 2

One Real Solution

Solve: x² - 6x + 9 = 0

Identifya = 1, b = -6, c = 9

Discriminantb² - 4ac = 36 - 36 = 0

Apply formulax = 6/2 = 3

Answer: x = 3 (repeated root)

Complex Solutions

Solve: x² + 2x + 5 = 0

Identifya = 1, b = 2, c = 5

Discriminantb² - 4ac = 4 - 20 = -16

Apply formulax = (-2 ± 4i) / 2

Answer: 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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