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Algebra · Calculus

Quadratic Equation Solver

Solve quadratic equations ax² + bx + c = 0 with comprehensive step-by-step math breakdowns, behavior presets, and real-time interactive plotting.

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⚡ Behavior Presets

3x² + 5x - 2 = 0

Live Equation Preview

Coefficient (a)3

Coefficient (b)5

Constant (c)-2

Roots & Discriminant

Discriminant (Δ)

Root x₁

0.3333 (1/3)

Root x₂

-2.0000

📈 Curve Properties

Vertex (h, k)(-0.83, -4.08)

Axis of Symmetryx = -0.83

Y-Intercept(0, -2)

DirectionOpens Up ▲

Visual Parabola Plot

Curve Roots

Scale: 1 unit = 20px

Detailed Mathematical Solution

Step 1

Equation coefficients: a = 3, b = 5, c = -2

Step 2

Discriminant Δ = b² - 4ac = (5)² - 4(3)(-2)

Step 3

Δ = 25 - (-24) = 49

Step 4

Since Δ > 0, equation has two distinct real roots: x = (-b ± √Δ) / 2a

Step 5

x = (-(5) ± √49) / (2 * 3)

Step 6

x = (-5 ± 7.0000) / 6

Step 7

x₁ = (-5 + 7.0000) / 6 = 0.3333 (1/3)

Step 8

x₂ = (-5 - 7.0000) / 6 = -2.0000

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Understanding Quadratic Equations

A quadratic equation is a second-order algebraic polynomial. The graph of a quadratic equation represents a symmetrical curve known as a parabola.

Standard Quadratic Formula

x = [ -b ± √(b² - 4ac) ] / 2a

What Vertex represents?

The vertex coordinate represents the absolute minimum point (if opens up, where a > 0) or maximum point (if opens down, where a < 0) of the parabola.

Common FAQs

What if the discriminant (Δ) is negative?

When Δ < 0, there are no real roots. Instead, the parabola curve sits entirely above or below the x-axis and has two complex conjugate solutions containing the imaginary unit i.

How is the Axis of Symmetry calculated?

The axis of symmetry represents a vertical mirror line dividing the parabola. Its equation is x = -b / 2a, which corresponds exactly to the x-coordinate of the vertex.

What does y-intercept represent?

The y-intercept represents the point where the curve intersects the vertical axis. It is obtained simply by setting x = 0 in the equation, yielding the value c.