Quadratic Formula Calculator

Solve quadratic equations of the form ax² + bx + c = 0, find real and complex roots, and understand how the discriminant describes different cases.

Enter Coefficients (a, b, c)

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Quadratic Equations and the Quadratic Formula

Quadratic equations model parabolic curves and many real‑world situations such as projectiles, optimization, and area problems.

This guide matches your other calculator pages, with boxes, examples, tables, pitfalls, and practice focused on the quadratic formula and discriminant.

What Is a Quadratic Equation?

A quadratic equation in one variable has the standard form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

Key pieces
  • a: Coefficient of x², controls how “wide” or “narrow” the parabola is.
  • b: Coefficient of x, affects the horizontal position of the vertex.
  • c: Constant term, gives the y‑intercept (value when x = 0).
Equation a b c
x² − 3x + 2 = 0 1 −3 2
2x² + 5x − 3 = 0 2 5 −3
−x² + 4x − 1 = 0 −1 4 −1
Example:
For 3x² − 6x + 1 = 0, a = 3, b = −6, c = 1.

The Quadratic Formula

The quadratic formula gives the solutions (roots) of ax² + bx + c = 0 directly in terms of a, b, and c.

Formula

x = [−b ± √(b² − 4ac)] / (2a)

Example 1: Solve x² − 3x + 2 = 0
a = 1, b = −3, c = 2.
Discriminant D = b² − 4ac = (−3)² − 4·1·2 = 9 − 8 = 1.
x = [3 ± √1] / 2 → x = (3 + 1)/2 = 2, or x = (3 − 1)/2 = 1.
Example 2: Solve 2x² + 5x − 3 = 0
a = 2, b = 5, c = −3.
D = 5² − 4·2·(−3) = 25 + 24 = 49; √D = 7.
x = [−5 ± 7]/(4) → x₁ = (−5 + 7)/4 = 0.5, x₂ = (−5 − 7)/4 = −3.

Discriminant and Types of Roots

The discriminant D = b² − 4ac determines how many real roots the quadratic has and whether they are distinct or repeated.

Cases
  • D > 0: Two distinct real roots.
  • D = 0: One real double root (repeated root).
  • D < 0: Two complex conjugate roots.
Equation D = b² − 4ac Type of roots
x² − 3x + 2 = 0 1 Two distinct real roots
x² − 4x + 4 = 0 0 One real double root
x² + 2x + 5 = 0 4 − 20 = −16 Two complex roots
Example:
For x² + 2x + 5 = 0, a = 1, b = 2, c = 5, D = 4 − 20 = −16 < 0, so the roots are complex.

Vertex, Axis of Symmetry, and Graph

The quadratic graph is a parabola, and the quadratic formula is closely related to its vertex and symmetry.

Key formulas

Axis of symmetry: x = −b / (2a)
Vertex: (−b / (2a), f(−b / (2a)))

Example:
For y = x² − 4x + 3, a = 1, b = −4, c = 3.
Axis: x = 4/2 = 2. Vertex x = 2, y = 2² − 4·2 + 3 = 4 − 8 + 3 = −1, so vertex is (2, −1).
Equation a Opens Axis
y = x² − 4x + 3 1 Upwards x = 2
y = −2x² + x + 1 −2 Downwards x = −1 / (4)

Applications of Quadratic Equations

Quadratic equations appear in physics, economics, geometry, and optimization whenever a relationship involves squares.

🎯 Projectile Motion

Height vs. time of a thrown object is quadratic, and roots give launch and landing times.

📈 Profit and Revenue

Quadratics model revenue or profit as functions of price or quantity, with the vertex giving optimal values.

📐 Geometry and Area

Problems about maximizing areas with fixed perimeters often lead to quadratic equations.

⚙️ Engineering & Robotics

Quadratics describe parabolic paths, sensor ranges, and kinematic equations in robotics projects.

Common Mistakes with the Quadratic Formula

Most errors come from sign mistakes, missing parentheses, or forgetting that a ≠ 0.

❌ MISTAKE 1: Forgetting parentheses around 2a
Writing −b ± √(b² − 4ac) / 2a without grouping changes the order of operations.
✅ Always divide the entire numerator by 2a: (−b ± √D) / (2a).
❌ MISTAKE 2: Using wrong signs for b or c
Copying coefficients incorrectly from the equation changes the discriminant and roots.
✅ Rewrite the equation in standard form ax² + bx + c = 0 first, then read off a, b, c carefully.
❌ MISTAKE 3: Treating linear equations as quadratic
If a = 0, the equation is not quadratic and the formula does not apply.
✅ For a = 0, solve the simpler linear equation bx + c = 0 instead.
❌ MISTAKE 4: Ignoring complex roots
When D < 0, roots are complex, not “no solution”.
✅ Write them as p ± qi using √(−D) = i√|D|.

Practice Problems with Solutions

Use these problems to practice; then verify your answers with the Quadratic Formula Calculator above.

Basic Level
Problem 1: Solve x² − 5x + 6 = 0.
Show Solution

a = 1, b = −5, c = 6; D = 25 − 24 = 1.
x = [5 ± 1]/2 → x = 3 or x = 2.

Problem 2: Solve 2x² + 3x − 2 = 0.
Show Solution

a = 2, b = 3, c = −2; D = 9 + 16 = 25.
x = [−3 ± 5]/4 → x = 0.5 or x = −2.

Intermediate Level
Problem 3: Solve x² + 4x + 4 = 0 and describe the roots.
Show Solution

a = 1, b = 4, c = 4; D = 16 − 16 = 0.
x = [−4 ± 0]/2 = −2 (double root).

Problem 4: Solve x² + 2x + 5 = 0 (complex roots).
Show Solution

a = 1, b = 2, c = 5; D = 4 − 20 = −16.
x = [−2 ± √(−16)] / 2 = [−2 ± 4i] / 2 = −1 ± 2i.

Advanced Level
Problem 5: A ball is thrown upward with height given by h(t) = −5t² + 20t + 3. When does it hit the ground (h = 0)?
Show Solution

Solve −5t² + 20t + 3 = 0 → 5t² − 20t − 3 = 0 (multiply by −1).
a = 5, b = −20, c = −3; D = 400 + 60 = 460.
t = [20 ± √460]/10 ≈ [20 ± 21.447]/10 → positive root ≈ 4.1447 s.

Problem 6: For y = 2x² − 8x + 1, find the x‑coordinates where y = 0 and the x‑coordinate of the vertex.
Show Solution

Solve 2x² − 8x + 1 = 0 → a = 2, b = −8, c = 1.
D = 64 − 8 = 56; x = [8 ± √56]/4 = [8 ± 2√14]/4 = 2 ± (√14)/2.
Vertex x = −b/(2a) = 8/4 = 2.