Quadratic equation 5x² + bx + c = 0

How the Quadratic Equation Calculator Works

Enter coefficients a, b, and c of the equation ax² + bx + c = 0. The calculator computes the discriminant (Δ = b² - 4ac), finds the roots using the quadratic formula x = (-b ± √Δ) / 2a, and determines the vertex of the parabola y = ax² + bx + c.

Quick start: Enter a=1, b=-5, c=6 and click Solve. Result: Δ = 1, roots x₁ = 2 and x₂ = 3, vertex V = (2.50, -0.25).

The Quadratic Formula and Discriminant

Every quadratic equation ax² + bx + c = 0 (where a ≠ 0) can be solved with the quadratic formula: x = (-b ± √Δ) / 2a. The discriminant Δ = b² - 4ac determines the nature and number of solutions.

Discriminant	Roots	Graph interpretation
Δ > 0	2 distinct real roots	Parabola crosses the x-axis at two points
Δ = 0	1 repeated root	Parabola is tangent to the x-axis (touches at one point)
Δ < 0	No real roots	Parabola does not intersect the x-axis (complex roots only)

Vertex of the Parabola

The vertex is the turning point of the parabola. If a > 0 the vertex is the minimum, if a < 0 it is the maximum. Coordinates: p = -b / (2a) and q = -Δ / (4a). The vertex form of the equation is y = a(x - p)² + q.

Practical Examples

Example 1: x² - 5x + 6 = 0 (a=1, b=-5, c=6)
Δ = 25 - 24 = 1 x₁ = 2, x₂ = 3, V = (2.50, -0.25)

Example 2: x² - 4x + 4 = 0 (a=1, b=-4, c=4)
Δ = 16 - 16 = 0 x₀ = 2 (repeated root), V = (2, 0)

Example 3: x² + 1 = 0 (a=1, b=0, c=1)
Δ = 0 - 4 = -4 no real roots (complex: x = ±i)

Example 4: 2x² + 3x - 2 = 0 (a=2, b=3, c=-2)
Δ = 9 + 16 = 25 x₁ = -2, x₂ = 0.5, V = (-0.75, -3.13)

Example 5: -x² + 6x - 5 = 0 (a=-1, b=6, c=-5)
Δ = 36 - 20 = 16 x₁ = 1, x₂ = 5, V = (3, 4) - maximum

FAQ

What is the discriminant and why does it matter?

The discriminant Δ = b² - 4ac tells you how many real solutions exist before you even solve the equation. Δ > 0 means two roots, Δ = 0 means one repeated root, Δ < 0 means no real roots (only complex). It is the expression under the square root in the quadratic formula.

What happens when a = 0?

If a = 0 the equation becomes linear: bx + c = 0, with solution x = -c/b. It is no longer quadratic. The calculator requires a ≠ 0.

What are complex roots?

When Δ < 0 the square root of a negative number appears. The roots become complex: x = (-b ± i√|Δ|) / 2a, where i is the imaginary unit (i² = -1). These roots always come in conjugate pairs.

How is the vertex useful?

The vertex gives the maximum or minimum value of the quadratic function. In optimization problems (maximizing profit, minimizing cost, finding peak height of a projectile) the vertex is the answer.

Where are quadratic equations used in real life?

Projectile motion (physics), profit maximization (economics), area optimization (geometry), signal processing (engineering), and pricing models (business). Any situation where a quantity depends on the square of a variable.

Can I use decimal coefficients?

Yes. The calculator accepts any real number for a, b, and c - integers, decimals, and negative values all work. For example a=0.5, b=-1.3, c=2.7 is valid.

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