What is a quadratic equation?

A quadratic equation has the form ax² + bx + c = 0 where a, b, c are constants and a ≠ 0. It’s called ‘quadratic’ because the highest power of x is 2 (Latin ‘quadratum’ = square). Quadratics appear constantly in physics (projectile motion, harmonic oscillators), engineering (structural analysis), economics (cost-revenue models), and everywhere two variables relate parabolically. The quadratic formula – x = (-b ± √(b² – 4ac)) / 2a – is one of the most important formulas in mathematics, allowing you to find the roots (values of x where the equation equals zero) for ANY quadratic equation. This calculator gives you the roots, discriminant (which determines the nature of solutions), and the vertex form of the parabola.

How to use this tool

1. Enter coefficients a, b, c — From ax² + bx + c = 0. a must be non-zero. Negative values allowed.
2. Read the equation — Tool displays your equation in standard form for verification.
3. Get the solutions — Root 1 and Root 2 with discriminant. Discriminant tells you the nature: positive = two real roots, zero = one repeated root, negative = complex roots.
4. See vertex — The vertex of the parabola (minimum or maximum point) shown as (h, k).

Quadratic formula

x = (-b ± √(b² – 4ac)) / 2a

The ± gives the two solutions.

Discriminant: Δ = b² – 4ac

• Δ > 0: Two distinct real roots
• Δ = 0: One real repeated root (double root)
• Δ < 0: Two complex conjugate roots (no real solutions)

Vertex form: y = a(x – h)² + k

• Vertex x-coordinate: h = -b / (2a)
• Vertex y-coordinate: k = c – b²/(4a) = a(h)² + b(h) + c

Worked example: Solve x² – 5x + 6 = 0 (so a=1, b=-5, c=6):

• Discriminant: (-5)² – 4(1)(6) = 25 – 24 = 1 (positive, so two real roots)
• x = (5 ± √1) / 2 = (5 ± 1) / 2
• x₁ = 6/2 = 3, x₂ = 4/2 = 2
• Vertex: h = 5/2 = 2.5, k = 6 – 25/4 = -1/4 = -0.25

Examples

• x² – 5x + 6 = 0: Roots 2 and 3 (factors easily as (x-2)(x-3) = 0)
• x² – 4 = 0: Roots ±2 (difference of squares)
• x² + 2x + 1 = 0: Roots -1 (repeated, since (x+1)² = 0)
• x² + 1 = 0: Roots ±i (complex – no real solutions)
• Projectile: h(t) = -5t² + 20t (height in m vs time t in sec): Roots t=0, t=4. Vertex at t=2, h=20m (max height)
• Profit: P(x) = -2x² + 100x – 500 (price-profit model): Max profit at x = 25, P = 750 (vertex)

Tips & best practices

• Try factoring first – if (x-2)(x-3) = 0, roots are 2 and 3 instantly (faster than quadratic formula)
• Discriminant of 0 means parabola TOUCHES x-axis at one point (vertex on axis)
• Vertex is min when a > 0 (parabola opens up), max when a < 0 (opens down)
• Sum of roots = -b/a; product of roots = c/a. Useful for verifying or constructing equations
• For projectile motion, the maximum height is at vertex; total time of flight is between the two real roots
• Optimization problems (maximize profit, minimize cost) often reduce to finding vertex of a parabola
• Complex roots come in conjugate pairs (a+bi and a-bi) – same real part, opposite imaginary parts

Limitations & notes

Calculator handles standard form ax² + bx + c = 0. For equations like ax² = bx + c, rearrange to standard form first. Doesn’t solve cubic (degree 3), quartic, or higher polynomials – those need different methods (Cardano’s formula, numerical methods). For systems of quadratic equations (2 equations, 2 unknowns), use simultaneous equation solvers.

Frequently Asked Questions

Why is it called ‘quadratic’ when there’s no quadrilateral?

From Latin ‘quadratum’ meaning ‘square’. The x² term involves squaring x. Nothing to do with quadrilaterals (4-sided figures). Many math terms come from Latin geometry roots – ‘cubic’ from cube, ‘quadratic’ from square.

When does a quadratic have no real solutions?

When discriminant b² – 4ac < 0. The parabola doesn’t cross the x-axis – it’s entirely above (if a>0) or below (if a<0). Solutions exist in complex numbers but not real numbers.

What’s the vertex of a parabola?

The minimum point if parabola opens upward (a>0), or maximum if opens downward (a<0). Coordinates: x = -b/(2a), y = c – b²/(4a). Important in optimization problems – the vertex IS the optimum value.

Can I solve x² + x = 1 with this?

Rearrange to standard form first: x² + x – 1 = 0. Now a=1, b=1, c=-1. Plug in. The ‘golden ratio’ relates to this equation: x = (1 + √5) / 2.

What’s special about the discriminant?

It’s the b² – 4ac part of the formula. Its SIGN determines root nature: positive (2 distinct real roots), zero (1 repeated real root), negative (complex roots). Useful even when you don’t need exact roots – just the type of solutions.

Can quadratics be factored?

Sometimes. If the roots are nice integers or simple fractions, factoring works: x² – 5x + 6 = (x-2)(x-3). For ‘ugly’ roots (irrational or complex), the quadratic formula is your friend. Practice both methods.

Where are quadratics used in real life?

Projectile motion (cannonball trajectory, basketball arc), area calculations (square garden), profit-loss models, structural engineering (parabolic arches), satellite dishes, optics, and acoustic design. Anywhere two variables relate to each other through squaring.

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