Solve a quadratic equation from its coefficients, see the discriminant and each step of the quadratic formula, or start from vertex form or factored form and convert it to standard form.
๐ Standard Form: axยฒ + bx + c = 0
๐ฏ Solutions
๐ Discriminant Analysis
๐ Step-by-Step Solution
๐ Additional Information
๐ก Understanding Quadratic Equations
- Discriminant (ฮ): bยฒ – 4ac determines the nature of roots
- ฮ > 0: Two distinct real roots
- ฮ = 0: One repeated real root
- ฮ < 0: Two complex conjugate roots
๐ฏ Vertex Form: a(x – h)ยฒ + k = 0
๐ฏ Solutions
๐ Standard Form Conversion
๐ Vertex Information
๐ก Vertex Form Benefits
- Directly shows the vertex (h, k) of the parabola
- Easy to identify the axis of symmetry: x = h
- Helps determine if parabola opens upward (a > 0) or downward (a < 0)
โ๏ธ Factored Form: a(x – rโ)(x – rโ) = 0
๐ Standard Form
๐ฏ Roots Analysis
๐ Key Properties
๐ก Factored Form Advantages
- Roots (x-intercepts) are immediately visible
- Easy to evaluate the function at specific points
- Useful for graphing and understanding the parabola’s behavior
What this quadratic solver does
- Standard Form: enter a, b and c for ax² + bx + c = 0 to get the roots, discriminant and worked steps.
- Vertex Form: enter a, h and k for a(x − h)² + k to see the standard form, the roots, the vertex, the axis of symmetry and whether the parabola opens up or down.
- Factored Form: enter a and the roots r₁ and r₂ to expand a(x − r₁)(x − r₂) into standard form.
How it is solved
The quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is the discriminant:
| Discriminant | Result |
|---|---|
| Greater than 0 | Two distinct real roots |
| Equal to 0 | One repeated real root |
| Less than 0 | Two complex conjugate roots |
Worked examples: x² − 3x + 2 = 0 has a discriminant of 1 and roots x = 2 and x = 1. x² + 2x + 5 = 0 has a discriminant of −16 and complex roots x = −1 ± 2i.
When a is positive the parabola opens upward and its vertex is the minimum point. When a is negative it opens downward and the vertex is the maximum.
Things to keep in mind
The coefficient a cannot be zero, because then the equation is not quadratic. Roots are shown to six decimal places. Values that differ from zero only by tiny rounding errors are treated as zero, so a repeated root is recognised.
Frequently asked questions
What does a negative discriminant mean?
The equation has no real solutions, but it has two complex solutions that are conjugates of each other.
What is the vertex?
It is the turning point of the parabola, at x = −b ÷ 2a in standard form.
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