How to use
Solve quadratic equations with exact roots, sequential explanations, a parabola graph and guided checks for every step.
Enter a quadratic equation using x² or x^2. Choose Show Me the Steps to see the full solution, or Let Me Solve It to check coefficients, discriminant, substitution and roots in order. Press Solve or Start solving after editing the equation. Fractions such as (1/2)x² and decimals such as 0.5x² are supported. An expression without an equals sign is treated as equal to zero.
Try a Similar Problem and New Problem generate a bounded practice equation in memory. Reset returns to 2x² + 5x − 3 = 0 and Show Me the Steps. Copy Solution exports the equation and full worked solution, excluding learner inputs. The default example, explanations, FAQs and native Why disclosures are visible without JavaScript.
The equation parser allows 256 characters, 128 tokens, eight nested numeric parentheses, 18 digits per number, coefficients with magnitude up to 1,000,000, and reduced denominators up to 1,000,000. It supports numeric coefficients multiplied by x or x², not functions, products of polynomial expressions or other variables. Coefficients and intermediate algebra use reduced BigInt fractions. No input executes as code.
There is no calculator storage, URL state, network calculation, analytics or account. Practice clears on refresh.
The quadratic formula
The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a). It solves ax² + bx + c = 0 whenever a ≠ 0. Completing the square produces this formula: after dividing by a and adding the square of b/(2a), take both square-root branches and isolate x.
Identifying a, b and c
First subtract the right-hand side from the left. For x² + 5x = 6, standard form is x² + 5x − 6 = 0, so a = 1, b = 5 and c = −6. The coefficient of a missing term is zero. In −x² + 4 = 0, a = −1, b = 0 and c = 4. Keep every sign with its coefficient.
The discriminant
D = b² − 4ac is the discriminant. Its sign classifies the roots before you simplify the formula. Positive D gives two distinct real roots, zero gives one repeated real root, and negative D gives two complex conjugate roots. The solver computes D with exact fractions.
Two real roots
When D > 0, adding and subtracting √D gives two distinct real values. A square discriminant can give rational roots; a nonsquare discriminant gives irrational roots for rational coefficients. We use deterministic + then − ordering in the centered form x = −b/(2a) ± √D/|2a|. This displays the larger real root first, regardless of the sign of a. Learner mode accepts either root order.
One repeated root
When D = 0, the square-root term vanishes. Both branches give x = −b/(2a). For x² + 4x + 4 = 0, D = 16 − 16 = 0 and the repeated root is x = −2. The parabola touches the x-axis at its vertex.
Complex roots
When D < 0, there are no real roots, but there are two complex conjugate roots. The imaginary unit i satisfies i² = −1, so √(−4) = 2i. For x² + 4x + 5 = 0, D = −4 and x = (−4 ± 2i)/2 = −2 ± i. The graph has no real x-intercepts.
Exact vs decimal answers
An exact fraction or radical retains the mathematical value. A decimal approximation is labeled Approximate and uses ≈. For x² − 2 = 0, x = ±√2 and x ≈ ±1.414214. Radicals simplify by extracting square factors: √12 = 2√3 and √8 = 2√2. Square extraction tests factors through 10,000 and checks a remaining perfect square; unusually large residual radicals may remain partially simplified.
You can review the precision of a decimal approximation as a separate formatting check. Rounding does not alter the exact root or increase mathematical accuracy, and it does not assign measurement uncertainty to exact coefficients.
Graph interpretation
The graph plots y = ax² + bx + c. Real roots are where the curve meets the x-axis. The vertex is (−b/(2a), c − b²/(4a)). Positive a opens upward; negative a opens downward. The viewport includes the roots, vertex and y-intercept with padding. Text values accompany every graph because this small SVG is an educational illustration, not a precision plotting package.
Limits and privacy
Use rational coefficients within the documented limits. General symbolic algebra, functions, other variables and higher powers are unsupported. Calculations and generated practice stay in memory in this tab. No calculator input is saved or sent over the network.
Worked examples
Two rational roots: 2x² + 5x − 3 = 0
- Identify a = 2, b = 5, c = −3.
- D = 5² − 4(2)(−3) = 25 + 24 = 49.
- x = (−5 ± √49) / 4.
- x = (−5 ± 7) / 4.
- x₁ = (−5 + 7)/4 = 1/2; x₂ = (−5 − 7)/4 = −3.
Final answer: x = 1/2 or x = −3. Approximate: x ≈ 0.5 or −3.
Irrational roots: x² − 3 = 0
a = 1, b = 0, c = −3. D = 12. Since √12 = 2√3, x = ±√3 ≈ ±1.732051.
Repeated root: x² + 4x + 4 = 0
D = 0; x = −4/2 = −2.
Complex roots: x² + 4x + 5 = 0
D = −4; x = (−4 ± 2i)/2 = −2 ± i. No real x-intercepts.
Quadratic formula FAQ
What is the quadratic formula?
It is x = (−b ± √(b² − 4ac))/(2a), used for ax² + bx + c = 0 with a ≠ 0.
How do I identify a, b and c?
Move all terms to the left, combine like terms, then read the signed coefficients of x², x and the constant.
What is the discriminant?
It is D = b² − 4ac, the expression inside the square root.
What does a positive discriminant mean?
There are two distinct real roots and two x-intercepts.
What happens when the discriminant is zero?
There is one repeated real root, located at the vertex.
What happens when the discriminant is negative?
There are no real roots and two complex conjugate roots.
Why does the quadratic formula use ±?
Taking a square root introduces positive and negative branches, which can produce two solutions.
When should I use the quadratic formula instead of factoring?
Use it when factors are difficult to recognize, or when you need a method that applies to any quadratic.
Why are some answers written with square roots?
A nonsquare discriminant produces irrational roots. Radicals preserve their exact values; rounded decimals do not.
Can every quadratic equation be solved with the quadratic formula?
Yes, with a ≠ 0, allowing complex roots when the discriminant is negative. This bounded tool accepts real rational coefficients in its supported notation.
Sources
Reviewed October 1, 2026. OpenStax explains standard form, the quadratic formula, discriminant classification and complex solutions. Parser limits, root ordering and practice generation are product choices. Sources do not endorse BuzCalculator.