Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with the quadratic formula. Get real or complex roots, the discriminant, and the vertex.
Independently verified for accuracy
Calculator by Toolsloft ↗- Roots
- x = 2 and x = 1
- Discriminant
- 1
- Vertex
- (1.5, -0.25)
Solve any quadratic equation of the form ax squared plus bx plus c equals zero. Enter the three coefficients and get the roots, whether they are two real numbers, one repeated root, or a complex pair, plus the discriminant and the vertex. Built for algebra homework and quick checks.
How this is calculated
Uses the quadratic formula, x = (-b plus or minus the square root of b squared minus 4ac) divided by 2a. The discriminant b squared minus 4ac sets the nature of the roots: positive gives two real roots, zero gives one repeated root, and negative gives a complex conjugate pair. The vertex sits at x = -b / 2a.
How to use
- Enter coefficient a (it cannot be zero).
- Enter coefficients b and c.
- Read the roots, the discriminant, and the vertex.
Examples
- Two real roots:
x^2 - 3x + 2 = 0 gives x = 2 and x = 1 - Complex roots:
x^2 + 1 = 0 gives x = 0 plus or minus 1i
FAQ
- What does the discriminant tell me?
- The discriminant is b squared minus 4ac. If it is positive there are two real roots, if it is zero there is one repeated root, and if it is negative the roots are a complex conjugate pair.
- Why must a not be zero?
- If a is zero the equation is linear, not quadratic, and the x squared term disappears. Use a linear equation solver in that case.
- How are complex roots shown?
- When the discriminant is negative, the tool reports the real part and the imaginary part, so the roots are the real part plus or minus the imaginary part times i.