Quadratic Formula Calculator and Quadratic Equation Solver — Step-by-Step

Paste an equation or enter a, b, and c for ax² + bx + c = 0 to get exact roots, rounded decimals, the discriminant, a graph, and full working.

Enter a quadratic

Accepts fractions, decimals, √ radicals, parentheses, implicit multiplication, and terms on either side. Use sqrt(2) or √2. Without =, the expression is set equal to zero.

Show steps using
Quick problem cases

Keyboard: arrow keys change input tabs; Ctrl/Cmd + Enter solves.

Solution

Enter an equation and choose Solve.
Your exact and decimal roots will appear here.

Step-by-step working

Steps will appear here.

Properties and alternate forms

Derived properties

Vertex
Awaiting solution
Axis of symmetry
Awaiting solution
Opening
Awaiting solution
Extremum
Awaiting solution
y-intercept
Awaiting solution

Alternate forms

Standard form

Awaiting solution

Vertex form

Awaiting solution

Factored form

Awaiting solution
Graph and intercepts
Solve a quadratic to render the graph.

The graph will show the parabola, intercepts, vertex, and axis of symmetry once solved.

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Calculator methodology and review

Developed and reviewed by
Starlight Robotics calculator team
Last reviewed
Formulas
Δ = b² − 4ac; x = (−b ± √Δ)/(2a); Vieta checks
Input limits
One variable x, polynomial degree ≤ 2, up to 500 characters; real numeric coefficients
Exactness and rounding
Integers, fractions, and finite decimals use rational arithmetic. Simplified exact roots are separated from approximations rounded to 10 places.
Representative tests
Real: x²−5x+6; repeated: x²−6x+9; irrational: 2x²−8x+5; complex: x²+4x+8; linear: 0x²+2x−4.

Understanding Quadratic Equations

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. Quadratics appear everywhere: projectile motion, optimization problems, area/geometry questions, and many modeling tasks in science and engineering.

The Quadratic Formula

Every quadratic can be solved using the quadratic formula:

x = [-b ± √(b2 - 4ac)] / (2a)

The expression under the square root, b² − 4ac, is called the discriminant (Δ). It determines the nature of the roots:

  • If Δ > 0 → two distinct real roots.
  • If Δ = 0 → one real repeated root.
  • If Δ < 0 → two complex conjugate roots.

How to use this calculator

  1. Paste an equation as written, or switch to Coefficients and enter a, b, and c.
  2. Choose the quadratic formula, factoring, or completing the square for the working.
  3. Click Solve Equation to compute the discriminant and roots.
  4. Review the exact answer first, then decimals, verification, properties, and graph.

Everything runs client-side in your browser, so your inputs never leave your device.

Identify a, b, and c

First move every term to one side. In x² + 4 = 5x, standard form is x² − 5x + 4 = 0, so a=1, b=−5, c=4. A missing term has coefficient zero: 2x²−7=0 has b=0.

Choose a solving method

Factor when simple factor pairs are visible. Complete the square to expose vertex form or derive the formula. Use the quadratic formula when factors are unclear, or roots are irrational or complex.

Read the graph and discriminant

Δ > 0 gives two x-intercepts, Δ = 0 gives one tangent intercept, and Δ < 0 gives none. The vertex is at x=−b/(2a); the sign of a determines whether the parabola opens up or down.

Check an answer

Substitute a root into both sides of the original equation. You can also check both roots with Vieta: r₁+r₂=−b/a and r₁r₂=c/a.

Fully worked examples for every root type

Two distinct real roots

x²−5x+6=0: Δ = 25−24 = 1. Thus x=(5±1)/2, so x=3 or x=2.

Vieta check: 3+2=5=−b/a and 3·2=6=c/a.

One repeated root

x²−6x+9=0: Δ = 36−36 = 0. Thus x=6/2=3, repeated.

Substitution: 3²−6·3+9=9−18+9=0.

Irrational radical roots

2x²−8x+5=0: Δ = 64−40 = 24. Then x=(8±√24)/4=(4±√6)/2.

Vieta check: the sum is 4=−b/a; the product is (16−6)/4=5/2=c/a.

Complex conjugate roots

x²+4x+8=0: Δ = 16−32 = −16. Then x=(−4±4i)/2=−2±2i.

Vieta check: the sum is −4; the product (−2+2i)(−2−2i)=4+4=8.

Fractional coefficients

(3/4)x²−(5/2)x+2=0. Multiply by 4: 3x²−10x+8=0. Δ = 100−96 = 4, so x=(10±2)/6: x=2 or x=4/3.

Vieta check: 2+4/3=10/3=−b/a and 2·4/3=8/3=c/a.

Practical input and interpretation guide

Move terms automatically

You may enter terms on either side, such as x²+4=5x. The solver subtracts the right side and shows the resulting standard form.

Equation input

Use zero for missing terms

For x²−9=0, enter a=1, b=0, c=−9 in coefficient mode—or simply omit the x term in equation mode.

Coefficients

Exact before approximate

(4+√6)/2 is exact. Its decimal is rounded, so use the exact form in symbolic work and the decimal for measurement or plotting.

Result reading

Roots and x-intercepts

Real roots are x-intercepts because they make the polynomial’s y-value zero. Complex roots do not appear as intercepts on a real-coordinate graph.

Graph meaning

Linear fallback

If a=0, the solver identifies the input as linear and solves bx+c=0 instead of applying a quadratic formula that would divide by zero.

Edge case

Quadratic Equation: FAQs

What is the quadratic formula?

x = [−b ± √(b² − 4ac)]/(2a). It solves ax²+bx+c=0 for x when a ≠ 0.

What does the discriminant (Δ) tell me?

Δ=b²−4ac. If Δ>0 there are two real roots; if Δ=0 there is one repeated real root; if Δ<0 there are two complex roots.

Are fractions and decimals allowed?

Yes. Use forms such as 3/4, -0.5, or (1/2)x². Finite decimals are kept as exact fractions for algebra; radicals in coefficients are accepted and evaluated.

What happens when a = 0?

The expression is not quadratic. The calculator solves the remaining linear equation bx+c=0, or reports an identity or contradiction when b is also zero.

How do I enter a missing b or c term?

Omit the term in equation mode, or enter 0 for its coefficient. For x²+5=0, b=0; for x²−3x=0, c=0.

Why can a quadratic have no real roots?

When b²−4ac is negative, the graph does not cross the x-axis and the two solutions are complex conjugates.

How are exact and decimal roots different?

An exact answer preserves fractions and simplified radicals. A decimal answer is a rounded approximation, shown to up to 10 decimal places.

When should I factor instead of using the quadratic formula?

Factoring is usually fastest when integer or simple rational factors are visible. The quadratic formula always works and is preferable for irrational or complex roots.

How can I verify the roots?

Substitute each root into the original equation, or check Vieta’s formulas: the sum is −b/a and the product is c/a.

Why are roots the x-intercepts?

A root makes y=ax²+bx+c equal zero. On the graph, y=0 is the x-axis, so real roots occur exactly where the parabola meets that axis.

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