What are complex roots of a quadratic equation?

When the discriminant D = b² - 4ac is negative, the quadratic equation has no real roots. Instead it has two complex conjugate roots of the form x = (-b/2a) ± (√|D|/2a)i, where i is the imaginary unit (√-1). Complex roots always come in conjugate pairs: if a+bi is a root, then a-bi is also a root. The parabola does not cross the x-axis when roots are complex.

What is vertex form of a quadratic equation?

Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. The vertex is the minimum point when a > 0 (parabola opens upward) or the maximum point when a < 0 (parabola opens downward). The vertex x-coordinate is h = -b/2a and y-coordinate is k = c - b²/(4a).

What is factored form of a quadratic equation?

Factored form is a(x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots. For example, x² - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0. Factored form only exists over real numbers when the discriminant is non-negative. When D < 0 (complex roots), the quadratic cannot be factored over real numbers.

How do I solve a quadratic equation step by step?

To solve ax² + bx + c = 0: Step 1: Identify a, b, and c. Step 2: Calculate the discriminant D = b² - 4ac. Step 3: If D ≥ 0, apply the quadratic formula x = (-b ± √D) / 2a to find the real roots. Step 4: If D < 0, the roots are complex: x = (-b/2a) ± (√|D|/2a)i. This solver shows every substitution step.

Can a quadratic equation have one solution?

Yes. When the discriminant D = b² - 4ac equals zero, the quadratic has exactly one solution (a repeated or double root): x = -b/2a. Geometrically, the parabola is tangent to the x-axis — it touches but does not cross it. This happens when the quadratic is a perfect square, for example x² + 2x + 1 = (x+1)² = 0 gives x = -1.

Quadratic Solver - LazyTools

Q: What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It solves any equation in the form ax² + bx + c = 0 where a ≠ 0. The formula gives two solutions (roots): x₁ using the + sign and x₂ using the - sign. When the discriminant (b² - 4ac) is negative, the roots are complex numbers.

Q: What is the discriminant?

The discriminant of a quadratic equation ax² + bx + c = 0 is D = b² - 4ac. It tells you how many real roots the equation has: D > 0 means two distinct real roots; D = 0 means one repeated real root (the parabola touches the x-axis at exactly one point); D < 0 means no real roots (two complex conjugate roots).

Q: What is factored form of a quadratic equation?

Factored form is a(x - r₁)(x - r₂) = 0, where r₁ and r₂ are the roots. For example, x² - 5x + 6 = 0 factors to (x - 2)(x - 3) = 0. Factored form only exists over real numbers when the discriminant is non-negative. When D < 0 (complex roots), the quadratic cannot be factored over real numbers.

Q: How do I solve a quadratic equation step by step?

To solve ax² + bx + c = 0: Step 1: Identify a, b, and c. Step 2: Calculate the discriminant D = b² - 4ac. Step 3: If D ≥ 0, apply the quadratic formula x = (-b ± √D) / 2a to find the real roots. Step 4: If D < 0, the roots are complex: x = (-b/2a) ± (√|D|/2a)i. This solver shows every substitution step.

Q: Can a quadratic equation have one solution?

Yes. When the discriminant D = b² - 4ac equals zero, the quadratic has exactly one solution (a repeated or double root): x = -b/2a. Geometrically, the parabola is tangent to the x-axis — it touches but does not cross it. This happens when the quadratic is a perfect square, for example x² + 2x + 1 = (x+1)² = 0 gives x = -1.

Q: Quadratic equation solver online with steps free

Enter a, b and c in the fields above and click Solve. The tool shows the discriminant, both roots (real or complex in a+bi form), standard form, vertex form, factored form, vertex coordinates, axis of symmetry and a parabola graph — with full step-by-step workings. Free, no account, instant results.

Key Features

More Than a Root Finder — Every Output a Student Needs

🌀

Complex roots in proper a+bi form — absent on most free tools — When the discriminant is negative (D < 0), the equation has no real roots. Most calculators either refuse to solve or display a vague error. This tool computes both complex roots in the form a ± bi where a = −b/2a and b = √|D|/2a — the mathematically correct complex conjugate pair. Essential for students studying A-level Maths, Further Maths and university-level algebra.

📈

Interactive parabola graph with vertex, roots and axis of symmetry — Auto-scaled SVG graph renders the parabola for all coefficient combinations. Root points (green), vertex (pink) and the axis of symmetry (purple dashed) are all labelled with coordinates. Parabola opens upward when a > 0 and downward when a < 0. For complex roots, the graph shows why the curve misses the x-axis entirely.

📏

Vertex form y = a(x−h)² + k — absent on all free tools — Vertex form makes the turning point immediately visible. The vertex (h, k) is extracted from the standard form coefficients: h = −b/2a, k = c − b²/4a. Vertex form is required for completing-the-square problems, graph transformations and optimisation problems.

🔢

Factored form a(x−r₁)(x−r₂) = 0 — When real roots exist, the factored form is shown with the actual root values. For a repeated root, it shows a(x−r)² = 0. For complex roots, a note explains that factoring over real numbers is impossible — an explanation absent on competing tools.

📋

Step-by-step quadratic formula with every substitution shown — Five numbered steps show: writing standard form, identifying coefficients, calculating the discriminant with full arithmetic, applying the formula with substituted values, and computing each root. Every number is substituted explicitly so students can follow and verify each stage of the working.

📐

All key properties in one view — Discriminant with interpretation, both roots, all three equation forms (standard, vertex, factored), vertex coordinates, axis of symmetry, y-intercept and parabola direction — all computed from a single solve. No need to run multiple separate calculators.

How To Use

Solve Any Quadratic in 3 Steps

Enter a, b and c — Type the three coefficients of your equation ax² + bx + c = 0. Use negative values for negative terms (e.g. x² − 5x + 6 is a=1, b=−5, c=6). Press Enter or click Solve.

Read the results — The discriminant tells you the case (two real, one repeated, or complex). Roots appear in the colour-coded boxes. Standard, vertex and factored forms are shown in the table below.

Check the graph and steps — The parabola graph shows vertex, roots and axis of symmetry visually. Scroll down for the step-by-step working showing every substitution into the quadratic formula.

Comparison

LazyTools vs Other Quadratic Solvers

Feature	LazyTools	Symbolab	Mathway	Desmos
Real roots with steps	✅ Yes	✅ Yes	⚠ Paid	⚠ No steps
Complex roots in a+bi form	✅ Yes	✅ Yes	⚠ Paid	❌ No
Vertex form output	✅ Yes	⚠ Partial	⚠ Paid	❌ No
Factored form output	✅ Yes	✅ Yes	⚠ Paid	❌ No
Parabola graph	✅ Yes	⚠ Separate	⚠ Paid	✅ Yes
Vertex coordinates	✅ Yes	✅ Yes	⚠ Paid	⚠ Manual
Free, no account	✅ Yes	⚠ Limited	❌ Paywalled	✅ Yes
All outputs in one view	✅ Yes	❌ Separate pages	❌ No	❌ No

Reference

Quadratic Formula Reference

Term	Formula	Description
Quadratic formula	x = (−b ± √D) / 2a	Solves any ax²+bx+c=0 where a≠0
Discriminant	D = b² − 4ac	Determines number and type of roots
Two real roots	D > 0	x₁ = (−b+𢆭)/2a, x₂ = (−b−𢆭)/2a
One repeated root	D = 0	x = −b/2a (double root)
Complex roots	D < 0	x = (−b/2a) ± (√\|D\|/2a)i
Vertex	(−b/2a, c−b²/4a)	Turning point of the parabola (h, k)
Axis of symmetry	x = −b/2a	Vertical line through the vertex
Vertex form	y = a(x−h)² + k	h = −b/2a, k = c − b²/4a
Factored form	a(x−r₁)(x−r₂) = 0	Only when D ≥ 0

Guide

Quadratic Equations — Complete Guide from Discriminant to Graph

What makes an equation quadratic?

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a, b and c are real numbers and a ≠ 0. The requirement that a ≠ 0 is essential — if a were zero, the x² term vanishes and the equation reduces to a linear equation bx + c = 0. The word "quadratic" comes from the Latin quadratus (square), referring to the x² term. Every quadratic equation has exactly two roots (counting multiplicity) in the complex numbers — this is a consequence of the Fundamental Theorem of Algebra.

The discriminant: the most important number in a quadratic

Before computing the roots, calculating D = b² − 4ac tells you everything about the nature of the solutions. When D > 0, the square root in the quadratic formula is real and positive, producing two distinct real roots that appear as x-intercepts on the parabola. When D = 0, the square root equals zero and both roots collapse to a single value x = −b/2a — the vertex of the parabola touches but does not cross the x-axis. When D < 0, the square root of a negative number is imaginary, producing two complex conjugate roots with no real x-intercepts.

Complex roots explained — what a+bi means

When D < 0, the quadratic formula gives x = (−b ± √D) / 2a. Since D is negative, √D = √(|D|) × i where i = √−1. The two roots are x₁ = −b/2a + (√|D|/2a)i and x₂ = −b/2a − (√|D|/2a)i — a complex conjugate pair sharing the same real part −b/2a but with opposite imaginary parts. Complex roots always come in conjugate pairs when a, b, c are real. This is why the vertex x-coordinate −b/2a equals the real part of both complex roots: it is still the axis of symmetry, it just does not correspond to an x-intercept.

Vertex form and completing the square

Vertex form y = a(x−h)² + k is obtained by completing the square on the standard form. Starting from ax² + bx + c, factor out a from the first two terms, complete the square inside the brackets, and simplify. The vertex (h, k) is immediately visible — h is the x-value of the turning point and k is the minimum (if a > 0) or maximum (if a < 0) value of the parabola. Vertex form is essential for graph transformations: y = a(x−h)² + k is the standard parabola y = x² scaled by a, shifted h units horizontally and k units vertically.

How to factorise a quadratic

Factorising means writing ax² + bx + c as a(x−r₁)(x−r₂) = 0 where r₁ and r₂ are the roots. The most reliable method for any quadratic is: (1) use the quadratic formula to find the roots, then (2) substitute into the factored form. For simple cases with integer roots, the "ac method" works: find two numbers that multiply to ac and add to b. For example x² − 5x + 6: ac = 6, and −2 × −3 = 6 with −2 + −3 = −5, giving (x−2)(x−3) = 0. When the discriminant is negative, factoring over real numbers is impossible — the quadratic is irreducible over the reals.

FAQ

Frequently Asked Questions

What is the quadratic formula?▾

The quadratic formula is x = (−b ± √(b²−4ac)) / 2a. It solves any equation ax²+bx+c=0 where a≠0. The ± gives two solutions: x₁ using + and x₂ using −. When the discriminant b²−4ac is negative, the roots are complex numbers.

What is the discriminant?▾

The discriminant is D = b² − 4ac. D > 0: two distinct real roots. D = 0: one repeated real root (parabola touches x-axis). D < 0: two complex conjugate roots (parabola misses x-axis entirely).

What are complex roots of a quadratic?▾

When D < 0, the roots are complex: x = (−b/2a) ± (√|D|/2a)i. They form a conjugate pair — same real part, opposite imaginary parts. This solver shows both roots in full a+bi form. The parabola does not cross the x-axis when roots are complex.

What is vertex form of a quadratic?▾

Vertex form is y = a(x−h)² + k where (h, k) is the vertex. h = −b/2a and k = c − b²/4a. The vertex is the minimum (a>0) or maximum (a<0) of the parabola.

How do I solve a quadratic step by step?▾

Step 1: Write in standard form ax²+bx+c=0. Step 2: Identify a, b, c. Step 3: Calculate D = b²−4ac. Step 4: Apply x = (−b ± √D) / 2a. Step 5: Compute both roots. This solver shows every substitution explicitly.

Can a quadratic have one solution?▾

Yes — when D = 0. The one solution is x = −b/2a, called a repeated or double root. The parabola is tangent to the x-axis at exactly one point. Example: x²+2x+1=0 gives x=−1 (double root).

Quadratic equation solver online with steps free▾

Enter a, b and c above and click Solve. See discriminant, roots (real or complex in a+bi form), standard form, vertex form, factored form, vertex, axis of symmetry and a parabola graph — all free with no account required.

What is factored form of a quadratic?▾

Factored form is a(x−r₁)(x−r₂)=0 where r₁ and r₂ are the roots. Example: x²−5x+6=(x−2)(x−3)=0. Only exists over real numbers when D≥0. When D<0 (complex roots), the quadratic cannot be factored over real numbers.

Related tools