Quadratic Equation Solver

What Is a Quadratic Equation?

A quadratic equation is any polynomial equation of the second degree, meaning the highest power of the unknown variable x is 2. The word "quadratic" comes from the Latin quadratus, meaning "square," because the equation involves squaring a variable.

Every quadratic equation can be written in standard form:

Where:

Coefficient	Name	Role	Constraint
a	Quadratic coefficient	Determines the width and direction of the parabola	a ≠ 0 (otherwise it's linear)
b	Linear coefficient	Shifts the vertex horizontally	Can be any real number
c	Constant term	The y-intercept of the parabola	Can be any real number

Examples of quadratic equations: x² − 5x + 6 = 0, 2x² + 3x − 2 = 0, x² + 1 = 0 (complex roots). An equation like 3x + 5 = 0 is not quadratic because the highest power is 1 (it's linear).

The Quadratic Formula

The quadratic formula provides a direct way to find the solutions (roots) of any quadratic equation. It is one of the most important formulas in algebra:

This formula works for every quadratic equation — whether the roots are rational, irrational, or complex. The "±" symbol means there are two solutions: one using addition and one using subtraction.

Derivation from Completing the Square

The quadratic formula isn't magic — it is derived by applying the completing the square technique to the general equation ax² + bx + c = 0:

1. Start: ax² + bx + c = 0
2. Divide by a: x² + (b/a)x + c/a = 0
3. Move constant: x² + (b/a)x = −c/a
4. Complete the square: Add (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = (b/2a)² − c/a
5. Factor left side: (x + b/2a)² = (b² − 4ac) / 4a²
6. Take square root: x + b/2a = ±√(b² − 4ac) / 2a
7. Solve for x: x = (−b ± √(b² − 4ac)) / 2a ✓

This derivation is a standard topic in Algebra I and Algebra II courses across the United States and appears on the SAT, ACT, and GRE math sections.

Understanding the Discriminant (Δ)

The expression under the square root — Δ = b² − 4ac — is called the discriminant. It determines the nature and number of solutions without actually solving the equation:

Discriminant Value	Number of Roots	Type of Roots	Graph Behavior
Δ > 0 (positive)	2	Two distinct real roots	Parabola crosses x-axis at two points
Δ = 0	1	One repeated (double) root	Parabola touches x-axis at vertex
Δ < 0 (negative)	0 real / 2 complex	Two complex conjugate roots	Parabola does not cross x-axis

Additionally, when Δ is a perfect square (like 1, 4, 9, 16, 25, …), the roots are rational numbers and the equation can be factored over the integers. Our calculator above automatically detects this and shows the exact fraction form and factored form when applicable.

Three Methods for Solving Quadratic Equations

There are three standard methods for solving quadratic equations. Each has advantages depending on the equation:

Method	Best For	Pros	Cons
Factoring	Simple integer roots	Fastest when applicable; no calculator needed	Only works when roots are rational
Quadratic Formula	Any quadratic equation	Universal — works for all cases including complex roots	More arithmetic steps
Completing the Square	Converting to vertex form	Reveals vertex; foundation for deriving the formula	Tedious for non-unit leading coefficients

Pro tip: Always check the discriminant first. If Δ is a perfect square, try factoring. If not, go straight to the quadratic formula. Our calculator uses the quadratic formula for all cases and automatically simplifies radicals and fractions.

Worked Examples

Example 1: Two Distinct Real Roots (Δ > 0)

Solve: x² − 5x + 6 = 0

1. Identify: a = 1, b = −5, c = 6
2. Discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1
3. Since Δ = 1 > 0 and is a perfect square → two rational roots
4. x = (5 ± √1) / 2 = (5 ± 1) / 2
5. x₁ = (5 + 1) / 2 = 3
6. x₂ = (5 − 1) / 2 = 2

Factored form: (x − 3)(x − 2) = 0. Verify: 3 × 2 = 6 ✓ and 3 + 2 = 5 ✓ (Vieta's formulas).

Example 2: One Repeated Root (Δ = 0)

Solve: x² − 6x + 9 = 0

1. Identify: a = 1, b = −6, c = 9
2. Discriminant: Δ = (−6)² − 4(1)(9) = 36 − 36 = 0
3. Since Δ = 0 → one repeated root
4. x = −(−6) / 2(1) = 6/2 = 3

Factored form: (x − 3)² = 0. The parabola touches the x-axis at its vertex (3, 0).

Example 3: Complex Conjugate Roots (Δ < 0)

Solve: x² + 2x + 5 = 0

1. Identify: a = 1, b = 2, c = 5
2. Discriminant: Δ = (2)² − 4(1)(5) = 4 − 20 = −16
3. Since Δ < 0 → two complex conjugate roots
4. √|Δ| = √16 = 4
5. Real part: −b/2a = −2/2 = −1
6. Imaginary part: ±4/2 = ±2i
7. x₁ = −1 + 2i, x₂ = −1 − 2i

The parabola y = x² + 2x + 5 has its vertex at (−1, 4) and never crosses the x-axis.

Example 4: Real-World Word Problem — Projectile Motion

A baseball is hit from a height of 3 feet with an initial upward velocity of 80 feet per second. Using the US standard gravitational acceleration (g = 32 ft/s²), the height equation is:

Question: When does the ball hit the ground? (h = 0)

1. Set h(t) = 0: −16t² + 80t + 3 = 0
2. a = −16, b = 80, c = 3
3. Δ = 80² − 4(−16)(3) = 6400 + 192 = 6592
4. √6592 = √(64 × 103) = 8√103 ≈ 81.20
5. t = (−80 ± 8√103) / (−32)
6. t₁ = (−80 + 81.20) / (−32) = −0.04 s (rejected, before hit)
7. t₂ = (−80 − 81.20) / (−32) ≈ 5.04 seconds

The ball hits the ground after approximately 5.04 seconds. Our calculator shows the exact form using simplified radicals — try entering a = −16, b = 80, c = 3.

Complex Roots & Imaginary Numbers

When the discriminant is negative (Δ < 0), the square root of a negative number is required. This introduces the imaginary unit:

A complex number has the form a + bi, where a is the real part and b is the imaginary part. Complex roots of quadratic equations always come in conjugate pairs: if a + bi is a root, then a − bi is also a root.

Complex numbers are not just mathematical abstractions — they are essential in electrical engineering (AC circuit analysis uses impedance as a complex number), signal processing (Fourier transforms), quantum mechanics, and control theory.

Vertex & Parabola Properties

Every quadratic equation y = ax² + bx + c graphs as a parabola. Key properties:

Property	Formula	Meaning
Vertex	(h, k) where h = −b/2a, k = c − b²/4a	The highest or lowest point of the parabola
Axis of Symmetry	x = −b/2a	The vertical line through the vertex
Direction	a > 0 → opens up; a < 0 → opens down	Determines if vertex is a minimum or maximum
Y-intercept	(0, c)	Where the parabola crosses the y-axis
Vertex Form	y = a(x − h)² + k	Alternative form revealing the vertex directly

Our calculator displays the vertex, axis of symmetry, y-intercept, and factored form (when roots are rational) for complete parabola analysis. For related geometry calculations, see our Area Calculator.

Real-World Applications of Quadratic Equations (USA Focus)

Quadratic equations appear throughout science, engineering, business, and everyday life:

Physics — Projectile Motion

In the United States, physics courses use the equation h(t) = −16t² + v₀t + h₀ (in feet, using g = 32 ft/s²) or h(t) = −4.9t² + v₀t + h₀ (in meters). This is the basis for calculating the trajectory of a thrown baseball, a launched rocket, or a basketball shot. The roots of the equation tell you when the object hits the ground.

Business — Profit Optimization

Revenue and profit functions are often quadratic: P(x) = −ax² + bx − c, where x is the quantity produced. The vertex gives the maximum profit point — the optimal quantity to produce. This is taught in MBA programs and business finance courses across the US.

Engineering & Architecture

Parabolic shapes appear in satellite dishes, headlight reflectors, suspension bridge cables, and the famous Gateway Arch in St. Louis (which is actually a catenary, closely related to a parabola). Engineers use quadratic equations to design these structures, calculate load distributions, and model stress curves.

Sports Analytics

In American football, the trajectory of a field goal kick follows a parabolic path. Coaches and analysts use quadratic models to determine the optimal launch angle and initial velocity for maximum distance or to clear the crossbar at 10 feet high.

The Golden Ratio — Derived from a Quadratic

One of mathematics' most famous numbers — the golden ratio φ (phi) — comes from solving a quadratic equation. If we want to split a line segment into two pieces where the ratio of the whole to the longer piece equals the ratio of the longer to the shorter:

The golden ratio appears in:

• Fibonacci sequence: The ratio of consecutive Fibonacci numbers approaches φ
• Regular pentagons: The diagonal-to-side ratio equals φ
• Art & architecture: The Parthenon, Leonardo da Vinci's works, and modern design
• Nature: Spiral patterns in sunflowers, pinecones, and nautilus shells

Try it in our calculator: enter a = 1, b = −1, c = −1. You'll get x₁ ≈ 1.618034 (the golden ratio) and x₂ ≈ −0.618034.

Vieta's Formulas — Sum & Product of Roots

Vieta's formulas provide a direct relationship between the roots and the coefficients of a quadratic equation, without actually solving for the roots:

Formula	Relationship	Example: x² − 5x + 6 = 0 (roots: 2, 3)
Sum of roots	x₁ + x₂ = −b/a	2 + 3 = 5 = −(−5)/1 ✓
Product of roots	x₁ · x₂ = c/a	2 × 3 = 6 = 6/1 ✓

These formulas are invaluable for checking your work and for constructing quadratic equations when you know the desired roots. Our calculator displays both Vieta's values automatically. For working with fractions in these calculations, use our Fraction Calculator.

Common Mistakes to Avoid

Students frequently make these errors when solving quadratic equations:

1. Forgetting the ± symbol: The formula gives two solutions. Always calculate both the + and − cases.
2. Sign errors with b: If b is negative, −b becomes positive. Be careful with double negatives: −(−5) = +5.
3. Dividing by 2a, not 2: The denominator is 2a, not just 2. For the equation 3x² + 6x + 2 = 0, the denominator is 2(3) = 6.
4. Forgetting that a ≠ 0: If a = 0, it's a linear equation, not quadratic. Solve with x = −c/b instead.
5. Not simplifying radicals: √50 should be simplified to 5√2. Our calculator handles this automatically.
6. Applying the formula to non-standard form: Always rearrange the equation to ax² + bx + c = 0 first. Move all terms to one side before identifying a, b, and c.

Use our Exponent Calculator when squaring the b coefficient, and our GCD Calculator to simplify fraction results.

Related Algebraic Concepts

• Percentage Calculator: Express discriminant ratios or root proportions as percentages
• Exponent Calculator: Compute powers needed for discriminant calculations (b²)
• Fraction Calculator: Simplify the fractional roots returned by the quadratic formula
• GCD Calculator: Find the GCD to reduce fractions and simplify radicals
• Standard Deviation Calculator: Analyze distributions related to quadratic models
• Glossary: Quadratic Equation: Full reference with discriminant table and solving methods compared