Quadratic Equation Solver
Solve ax² + bx + c = 0 instantly with full step-by-step solution.
Equation preview
x² − 5x + 6 = 0
Coefficient of x². Must not be 0.
Coefficient of x.
Constant term.
Solution x₁
(5 + 1) ÷ 2
Decimal: 3
Solution x₂
(5 − 1) ÷ 2
Decimal: 2
Discriminant
1
Two real roots
Root type
Two real roots
Vertex
(2.5, -0.25)
Turning point of the parabola.
Axis of symmetry
x = 2.5
Y-intercept
(0, 6)
Factored form
(x − 3)(x − 2)
Formula substitution
x = (-b ± √(b² − 4ac)) ÷ 2a
x = (5 ± √1) ÷ 2
Quadratic Formulas
Standard Form
ax² + bx + c = 0
Quadratic Formula
x = (-b ± √(b² − 4ac)) ÷ 2a
Discriminant
D = b² − 4ac
Axis of Symmetry
x = -b ÷ 2a
Vertex
xᵥ = -b ÷ 2a, yᵥ = f(xᵥ)
Vertex Form
y = a(x − h)² + k
Factored Form
a(x − r₁)(x − r₂)
Variable Explanations
a
Coefficient of x².
b
Coefficient of x.
c
Constant term.
x
Unknown variable.
D
Discriminant.
r₁ and r₂
Roots or solutions.
h and k
Vertex coordinates.
±
Plus or minus.
Parabola and Graph Meaning
This is an explanatory sketch, not a precise graph. A quadratic graph is a parabola. Real roots are x-intercepts, the vertex is the turning point, and the parabola opens upward when a is positive.
Worked Examples
x² − 5x + 6 = 0
Coefficients: a = 1, b = -5, c = 6
Discriminant: D = 25 − 24 = 1
Substitution: x = (5 ± √1) ÷ 2
Roots: x = 3 and x = 2
Two real roots.
x² + 2x + 1 = 0
Coefficients: a = 1, b = 2, c = 1
Discriminant: D = 4 − 4 = 0
Substitution: x = (-2 ± 0) ÷ 2
Roots: x = -1
One repeated root.
x² + 4x + 5 = 0
Coefficients: a = 1, b = 4, c = 5
Discriminant: D = 16 − 20 = -4
Substitution: x = (-4 ± √-4) ÷ 2
Roots: x = -2 ± i
Two complex roots.
2x² − 3x − 2 = 0
Coefficients: a = 2, b = -3, c = -2
Discriminant: D = 9 + 16 = 25
Substitution: x = (3 ± 5) ÷ 4
Roots: x = 2 and x = -0.5
Two real roots.
Find root type
Coefficients: Use D = b² − 4ac
Discriminant: D > 0, D = 0, or D < 0
Substitution: Compare D to 0
Roots: Root type follows D
The discriminant controls the type.
Find vertex
Coefficients: xᵥ = -b ÷ 2a
Discriminant: Substitute xᵥ into f(x)
Substitution: yᵥ = f(xᵥ)
Roots: Vertex = (h, k)
The vertex is the turning point.
Factorable example
Coefficients: x² − 7x + 12 = 0
Discriminant: D = 1
Substitution: (x − 3)(x − 4)
Roots: x = 3 and x = 4
Factoring works neatly.
Non-factorable example
Coefficients: x² + x − 1 = 0
Discriminant: D = 5
Substitution: x = (-1 ± √5) ÷ 2
Roots: Approx. 0.618 and -1.618
Use the formula.
Discriminant and Root Types
D > 0
Two real roots. The graph crosses the x-axis twice.
D = 0
One repeated real root. The graph touches the x-axis once.
D < 0
Two complex roots. The graph has no real x-intercepts.
Factoring vs Formula vs Completing the Square
Factoring
Fast when the quadratic splits neatly into simple factors.
Quadratic formula
Works for every quadratic equation.
Completing the square
Useful for vertex form and formula derivations.
Graphing
Helps interpret roots, vertex, and x-intercepts.
Common Quadratic Equation Mistakes
Understanding Your Results
Roots
x-values that make the equation equal zero.
Discriminant
Tells the number and type of roots.
Vertex
Turning point of the parabola.
Axis of symmetry
Vertical line through the vertex.
Y-intercept
Where the parabola crosses the y-axis.
Factored form
Product form showing roots when factorable.
Frequently Asked Questions
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