Quadratic Equation Solver | DigitHelm
Solve quadratic equations (ax² + bx + c = 0) and find both real and complex roots.
Solve ax² + bx + c = 0, enter coefficients below
All calculations run live in your browser using the quadratic formula.
What Is the Quadratic Equation Solver | DigitHelm?
A quadratic equation has the form ax² + bx + c = 0 where a ≠ 0. The quadratic formula always produces the correct roots, real or complex, regardless of the discriminant. This solver also computes the vertex, axis of symmetry, vertex form, and Vieta's formulas for sum and product of roots.
- ›The discriminant Δ = b² − 4ac determines how many real roots exist
- ›The vertex (h, k) gives the parabola's minimum (opens up) or maximum (opens down)
- ›Vertex form a(x − h)² + k is useful for graphing and transformations
- ›Vieta's formulas: x₁ + x₂ = −b/a and x₁ · x₂ = c/a
Formula
ax² + bx + c = 0
x = (−b ± √(b² − 4ac)) / 2a
Parameters:
- ›a, coefficient of x² (must be non-zero)
- ›b, coefficient of x
- ›c, constant term
- ›Δ = b² − 4ac, discriminant (determines root type)
Δ > 0
Two distinct real roots
Δ = 0
One repeated real root
Δ < 0
Two complex roots
How to Use
- 1Enter the coefficients a, b, and c for the equation ax² + bx + c = 0
- 2Coefficient a must be non-zero (if a = 0 it becomes a linear equation)
- 3Click Solve or press Enter in any field
- 4The discriminant is computed first to determine root type
- 5Roots, vertex, vertex form, and step-by-step solution are displayed
- 6Use the Clear button to reset all inputs
Example Calculation
Solve x² − 5x + 6 = 0 (a=1, b=−5, c=6):
√Δ = 1
x₁ = (5 + 1) / 2 = 3
x₂ = (5 − 1) / 2 = 2
Vertex: x = −(−5)/(2×1) = 2.5 → y = 6 − 25/4 = −0.25
Vertex form: (x − 2.5)² − 0.25
Complex roots example
For x² + 2x + 5 = 0: Δ = 4 − 20 = −16 < 0
x = (−2 ± √(−16)) / 2 = −1 ± 2i
Roots: x₁ = −1 + 2i, x₂ = −1 − 2i
Understanding Quadratic Equation | DigitHelm
Understanding Quadratic Equations
A quadratic equation is any polynomial equation of degree 2. The general form ax² + bx + c = 0 represents a parabola when graphed. The solutions (roots) are the x-intercepts of that parabola. Al-Khwarizmi described systematic methods for solving quadratics in his 9th-century treatise, the work that gave algebra its name.
- ›The graph is a parabola, symmetrical about the axis x = −b/(2a)
- ›The vertex is the highest or lowest point depending on the sign of a
- ›Roots are where the parabola intersects the x-axis (y = 0)
- ›The y-intercept is always at (0, c)
Discriminant Summary
| Discriminant | Root type | Parabola | Example |
|---|---|---|---|
| Δ > 0 (perfect sq) | Two rational | Crosses x-axis twice | x²−5x+6=0 → 2, 3 |
| Δ > 0 (not perf. sq) | Two irrational | Crosses x-axis twice | x²−3=0 → ±√3 |
| Δ = 0 | One double root | Tangent to x-axis | x²−2x+1=0 → 1, 1 |
| Δ < 0 | Two complex | Above or below axis | x²+1=0 → ±i |
Frequently Asked Questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. It is derived by completing the square on the general quadratic and always produces the correct roots, real or complex.
- ›The ± gives two solutions (or one when Δ = 0)
- ›It works for all real coefficients a, b, c (a ≠ 0)
- ›First documented in Indian mathematics around 1100 AD (Bhaskara II)
- ›Preferred over factoring when roots are not obvious integers
What does the discriminant tell me?
The discriminant is the expression under the square root sign: Δ = b² − 4ac. Its sign completely determines the nature of the roots before you even compute them.
- ›Δ > 0: two distinct real roots, the parabola crosses the x-axis twice
- ›Δ = 0: one repeated (double) root, the parabola is tangent to the x-axis
- ›Δ < 0: two complex conjugate roots, the parabola does not cross the x-axis
- ›Δ is a perfect square when the roots are rational
What is vertex form?
Vertex form a(x − h)² + k directly encodes the vertex (h, k) and the direction the parabola opens. Converting to vertex form is equivalent to completing the square.
- ›h = −b/(2a) is the x-coordinate of the vertex
- ›k = c − b²/(4a) is the y-coordinate (minimum or maximum)
- ›If a > 0, the parabola opens upward and (h, k) is a minimum
- ›If a < 0, the parabola opens downward and (h, k) is a maximum
What are Vieta's formulas?
Named after François Viète (1540–1603), Vieta's formulas provide a direct relationship between the roots and the coefficients without solving the equation.
- ›Sum of roots: x₁ + x₂ = −b/a
- ›Product of roots: x₁ · x₂ = c/a
- ›Useful for checking your solution: verify sum and product match
- ›Extend to higher-degree polynomials (Newton's identities)
Can a quadratic have no real solutions?
When Δ < 0, the square root of a negative number appears, producing complex (imaginary) roots. The parabola does not cross or touch the x-axis at all.
- ›Complex roots always come in conjugate pairs: a + bi and a − bi
- ›The real part = −b/(2a) is the axis of symmetry
- ›The imaginary part = √(−Δ)/(2a)
- ›Complex roots still satisfy Vieta's formulas for sum and product
How do I solve a quadratic by factoring?
Factoring is fast when the roots are integers or simple fractions, but the quadratic formula always works. The calculator uses the formula to handle all cases.
- ›Factor x² − 5x + 6: find two numbers multiplying to 6, adding to −5 → (x−2)(x−3)
- ›If Δ is not a perfect square, roots are irrational, use the formula
- ›If Δ < 0, factoring over the reals is impossible, use the formula for complex roots
- ›Completing the square is another algebraic method that derives the formula
What real-world problems use quadratic equations?
Quadratic equations appear whenever a squared relationship exists. They are foundational in physics, economics, and engineering.
- ›Projectile motion: height h(t) = −½gt² + v₀t + h₀
- ›Area problems: "a rectangle has area 24 and perimeter 20, find dimensions"
- ›Business: maximum profit occurs at the vertex of a quadratic cost function
- ›Optics: lens equation and focal-length calculations
- ›Electrical: resonance frequency in LC circuits