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Quadratic Equation Calculator

Solve ax² + bx + c = 0

Roots

3, 2

Discriminant

1.00

Type

2 Real

Vertex

(2.5, -0.3)

x² - 5x + 6 = 0

Solutions (Roots)

x₁

3

x₂

2

Two distinct real roots

Discriminant (Δ = b² - 4ac)

1.00

Δ > 0 → Two real roots

Parabola Properties

Vertex

(2.50, -0.25)

Axis of Symmetry

x = 2.50

Y-Intercept

(0, 6)

Opens

↑ Upward

Solution Steps

1. Equation: x² - 5x + 6 = 0

2. Using quadratic formula: x = (-b ± √(b²-4ac)) / 2a

3. a = 1, b = -5, c = 6

4. Discriminant: Δ = (-5)² - 4(1)(6) = 1.00

5. x = (-(-5) ± √1.00) / (2 × 1)

6. x₁ = 3, x₂ = 2

Parabola Shape

(2.5, -0.3)x₁x₂
VertexRoots

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Formulas Used

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

The quadratic formula finds the roots of any equation in the form ax² + bx + c = 0.

Where:

a= Coefficient of x² (must not be zero)
b= Coefficient of x
c= Constant term

Discriminant

Δ = b² - 4ac

The discriminant determines the nature of the roots: Δ > 0 gives two real roots, Δ = 0 gives one repeated root, Δ < 0 gives complex roots.

Where:

a= Coefficient of x²
b= Coefficient of x
c= Constant term

Vertex of Parabola

Vertex = (-b/(2a), a(-b/(2a))² + b(-b/(2a)) + c)

The vertex is the minimum or maximum point of the parabola.

Where:

-b/(2a)= The x-coordinate of the vertex (axis of symmetry)

Example Calculations

1Solve x² - 5x + 6 = 0

Inputs

a (x²)1
b (x)-5
c (const)6

Result

Rootsx₁ = 3, x₂ = 2
Discriminant1.00
Vertex(2.50, -0.25)
Root TypeTwo distinct real roots

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1. Since Δ > 0, there are two real roots. x₁ = (5 + √1) / 2 = 3, x₂ = (5 - √1) / 2 = 2. Vertex at x = 5/2 = 2.50, y = -0.25.

2Solve 2x² + 4x - 6 = 0

Inputs

a (x²)2
b (x)4
c (const)-6

Result

Rootsx₁ = 1, x₂ = -3
Discriminant64.00
Vertex(-1.00, -8.00)
Root TypeTwo distinct real roots

Discriminant = 4² - 4(2)(-6) = 16 + 48 = 64. x₁ = (-4 + √64) / 4 = (-4 + 8) / 4 = 1. x₂ = (-4 - 8) / 4 = -3. Vertex at x = -4/4 = -1.00, y = 2(1) + 4(-1) - 6 = -8.00.

3Solve x² + 4x + 4 = 0 (Repeated Root)

Inputs

a (x²)1
b (x)4
c (const)4

Result

Rootsx₁ = -2, x₂ = -2
Discriminant0.00
Vertex(-2.00, 0.00)
Root TypeOne repeated real root

Discriminant = 4² - 4(1)(4) = 16 - 16 = 0. Since Δ = 0, there is one repeated root. x = -4 / (2×1) = -2. The vertex is at (-2, 0), meaning the parabola just touches the x-axis.

4Solve x² + x + 1 = 0 (Complex Roots)

Inputs

a (x²)1
b (x)1
c (const)1

Result

Rootsx₁ = -0.5 + 0.866i, x₂ = -0.5 - 0.866i
Discriminant-3.00
Vertex(-0.50, 0.75)
Root TypeTwo complex conjugate roots

Discriminant = 1² - 4(1)(1) = 1 - 4 = -3. Since Δ < 0, the roots are complex. Real part = -1/(2×1) = -0.5. Imaginary part = √3/2 ≈ 0.866. The parabola does not cross the x-axis.

Frequently Asked Questions

Q

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It finds the roots (solutions) of any quadratic equation in the form ax² + bx + c = 0. This universal formula works for all quadratic equations, whether the roots are real or complex.

  • Works for any quadratic equation ax² + bx + c = 0
  • The ± sign means there are two solutions (x₁ and x₂)
  • a must not equal zero (otherwise it is linear)
  • Derived by completing the square on the general form
  • Also called the "abc formula" in some countries
Q

What does the discriminant tell us?

The discriminant (b² - 4ac) indicates the nature of the roots: positive means two real roots, zero means one real root (repeated), and negative means two complex conjugate roots.

  • Δ > 0: Two distinct real roots (parabola crosses x-axis twice)
  • Δ = 0: One repeated real root (parabola touches x-axis once)
  • Δ < 0: Two complex conjugate roots (parabola does not cross x-axis)
  • A larger Δ means the roots are further apart
  • Δ is always b² - 4ac, regardless of the sign of a
Discriminant (Δ)Number of RootsRoot TypeGraph Behavior
Δ > 02Two distinct real rootsCrosses x-axis twice
Δ = 01One repeated real rootTouches x-axis at vertex
Δ < 00 real (2 complex)Complex conjugate pairDoes not cross x-axis
Q

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. For y = ax² + bx + c, the vertex x-coordinate is -b/(2a), and it represents either the maximum (if a < 0) or minimum (if a > 0) value.

  • Vertex x-coordinate: x = -b/(2a)
  • Vertex y-coordinate: substitute x back into ax² + bx + c
  • If a > 0, vertex is the minimum point
  • If a < 0, vertex is the maximum point
  • The axis of symmetry passes through the vertex at x = -b/(2a)
Q

How do I know if the parabola opens up or down?

If the coefficient "a" is positive, the parabola opens upward (U-shape). If "a" is negative, the parabola opens downward (∩-shape). The magnitude of "a" controls how wide or narrow the parabola is.

  • a > 0: Opens upward (U-shape), has a minimum at the vertex
  • a < 0: Opens downward (∩-shape), has a maximum at the vertex
  • |a| > 1: Narrower parabola (steeper sides)
  • |a| < 1: Wider parabola (flatter sides)
  • |a| = 1: Standard-width parabola
Q

What are the solutions for common quadratic equations?

Many quadratic equations have clean integer or simple fraction roots. Recognizing common patterns like perfect square trinomials (x²+6x+9=0) and difference of squares (x²-9=0) helps solve equations faster without the full quadratic formula.

  • x² - 4 = 0 is a "difference of squares" → roots are ±2
  • x² - 6x + 9 = 0 is a "perfect square trinomial" → double root at 3
  • x² + 1 = 0 has no real roots (complex: ±i)
  • Factor when possible: x² + 5x + 6 = (x+2)(x+3)
  • Use quadratic formula when factoring is not obvious
Equationa, b, cDiscriminantRoots (x₁, x₂)
x² + 5x + 6 = 01, 5, 61-2, -3
x² - 4 = 01, 0, -4162, -2
2x² + 3x - 2 = 02, 3, -2250.5, -2
x² - 6x + 9 = 01, -6, 903, 3
x² + 1 = 01, 0, 1-4i, -i
x² - x - 6 = 01, -1, -6253, -2
Q

What are the different methods to solve quadratic equations?

There are four main methods: factoring, completing the square, the quadratic formula, and graphing. Factoring is fastest for simple equations. The quadratic formula always works. Completing the square is useful for deriving vertex form.

  • Factoring: Rewrite as (x - r₁)(x - r₂) = 0
  • Quadratic formula: Plug a, b, c directly
  • Completing the square: Rewrite as a(x - h)² + k = 0
  • Graphing: Plot y = ax² + bx + c and find x-intercepts
  • Always verify by substituting roots back into the original equation
MethodBest ForDifficultyAlways Works?
FactoringInteger roots, simple equationsEasyNo
Quadratic FormulaAny quadratic equationMediumYes
Completing the SquareConverting to vertex formMedium-HardYes
GraphingVisualizing rootsEasy (with tools)Approximate only

Understanding Quadratic Equations

1

The Quadratic Formula Step by Step

x = (−b ± √(b² − 4ac)) / 2a solves any equation of the form ax² + bx + c = 0 in a single computation. For x² − 5x + 6 = 0: a=1, b=−5, c=6. Discriminant = 25 − 24 = 1. Roots: x = (5 ± 1)/2, giving x₁ = 3 and x₂ = 2. Verification: 3² − 15 + 6 = 0 and 2² − 10 + 6 = 0.

The formula is derived by completing the square on the general form. Starting with ax² + bx + c = 0, divide by a, move c/a to the right, add (b/2a)² to both sides, factor the left as a perfect square, and solve for x. This derivation appears in virtually every algebra textbook and was known to Babylonian mathematicians around 2000 BCE, though they expressed solutions geometrically rather than algebraically.

The ± symbol produces two solutions because every parabola (except at the vertex) intersects a horizontal line at zero or two points. The expression under the square root — the discriminant b² − 4ac — determines whether those intersection points are real, coincident, or complex. The Square Root Calculator can verify the √(b² − 4ac) step for hand calculations.

Discriminant Determines Root TypeΔ > 0Two real rootsx₂ ≠ x₁Δ = 0One repeated rootx₁ = x₂Δ < 0Two complex rootsa ± biNo x-interceptsx = (−b ± √Δ) / 2awhere Δ = b² − 4ac2 real roots (Δ>0)1 repeated (Δ=0)2 complex (Δ<0)
2

The Discriminant and Root Classification

The discriminant Δ = b² − 4ac classifies roots before computing them. For 2x² + 4x − 6 = 0: Δ = 16 + 48 = 64 > 0, so two distinct real roots exist (x = 1 and x = −3). For x² + 4x + 4 = 0: Δ = 16 − 16 = 0, giving one repeated root (x = −2). For x² + x + 1 = 0: Δ = 1 − 4 = −3 < 0, producing complex roots (−0.5 ± 0.866i).

A perfect square discriminant (1, 4, 9, 16, 25...) means the roots are rational, which means the quadratic can be factored over the integers. Δ = 25 for x² + 5x + 6 = 0 gives roots −2 and −3, and the factored form is (x+2)(x+3). When Δ is positive but not a perfect square, roots are irrational: for x² − 2 = 0, Δ = 8, giving x = ±√2.

The discriminant also determines the parabola’s relationship to the x-axis. Δ > 0: the parabola crosses the axis at two points. Δ = 0: the vertex touches the axis (tangent). Δ < 0: the parabola floats entirely above (a > 0) or below (a < 0) the axis. The Logarithm Calculator can help verify solutions when quadratic equations arise from exponential or logarithmic equations.

The discriminant classifies roots without computing them fully
EquationΔ = b²−4acRoot TypeRoots
x² − 5x + 6 = 01 (perfect square)Two rational3, 2
x² − 2 = 08 (positive)Two irrational±√2
x² − 6x + 9 = 00One repeated3
x² + x + 1 = 0−3 (negative)Two complex−0.5 ± 0.866i
3

Vertex, Axis of Symmetry, and Optimization

The vertex of y = ax² + bx + c is at x = −b/(2a), which is the axis of symmetry. The y-coordinate is found by substituting back. For y = 2x² + 4x − 6: vertex x = −4/4 = −1, vertex y = 2(1) + 4(−1) − 6 = −8. This vertex (−1, −8) is the minimum because a = 2 > 0.

Vertex form y = a(x − h)² + k gives the vertex directly as (h, k). Converting from standard form: complete the square. For y = x² − 6x + 5: y = (x² − 6x + 9) − 9 + 5 = (x − 3)² − 4, so vertex = (3, −4). The value a = 1 > 0 confirms this is the minimum.

Real-world optimization problems frequently reduce to finding vertices. A ball thrown at 20 m/s at 45° follows h(t) = −4.9t² + 14.14t + 1.5 (height in meters). Maximum height occurs at t = −14.14/(2 × −4.9) = 1.44 seconds, giving h = 11.7 meters. Revenue models (R = price × quantity, where quantity decreases linearly with price) also produce quadratic functions whose vertex gives the profit-maximizing price.

Vertex Form: y = a(x−h)² + kExample: y = (x−3)² − 4Vertex (3, −4)x=1x=5axis: x=3Vertex (min/max)Roots (x-intercepts)Parabola (a>0: opens up)

When a > 0, the vertex is the minimum point. When a < 0, the vertex is the maximum. The axis of symmetry x = −b/(2a) always passes through the vertex and midway between the two roots (if they exist).

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Last Updated: Mar 26, 2026

This calculator is provided for informational and educational purposes only. Results are estimates and should not be considered professional financial, medical, legal, or other advice. Always consult a qualified professional before making important decisions. UseCalcPro is not responsible for any actions taken based on calculator results.

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