Free algebra calculator
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with the quadratic formula, including real roots, repeated roots, complex roots, discriminant, vertex, and axis of symmetry.
- Real and complex roots
- Discriminant result
- Step-by-step work
- Vertex and axis
Solution
Roots and discriminant
x^2 - 3x + 2 = 0x = 2
Enter a, b, and c to solve the equation.
- Root 1
- x = 2
- Root 2
- x = 1
- Discriminant
- 1
- Root type
- Two real roots
- Axis of symmetry
- x = 1.5
- Vertex
- (1.5, -0.25)
For homework or tests, show the exact form your teacher expects. This calculator displays practical decimal results and formula steps.
Formula work
Step-by-step solution
The calculator breaks the answer into the formula setup, discriminant, and final root calculation.
How the quadratic formula works
The quadratic formula solves equations in the form ax^2 + bx + c = 0. The coefficients a, b, and c are substituted into x = (-b +/- sqrt(b^2 - 4ac)) / 2a. This is useful when factoring is difficult, when the roots are irrational, or when you want a reliable way to check algebra work.
The value under the square root, b^2 - 4ac, is called the discriminant. It controls the kind of answer the equation has before you even finish solving.
What the discriminant tells you
If the discriminant is positive, the equation has two real roots. If it is zero, the equation has one repeated real root. If it is negative, the equation has two complex roots because the square root of a negative number requires i.
Students often use the discriminant as a quick answer check. For example, if a graph never crosses the x-axis, the discriminant should be negative and the roots should be complex.
Why the vertex and axis are included
The roots show where the parabola crosses the x-axis, but the vertex shows the high or low point of the graph. The axis of symmetry is the vertical line through that vertex. Together, these values make it easier to connect equation solving with graphing.
The axis of symmetry is x = -b / 2a. After finding that x value, the calculator substitutes it back into ax^2 + bx + c to estimate the vertex point.
Common quadratic formula mistakes
The most common mistakes are forgetting parentheses around negative b values, dropping the plus-or-minus sign, and dividing only part of the numerator by 2a. Another common issue is using the formula when a is zero. If a is zero, the equation is linear and should be solved as bx + c = 0.
When you are checking homework, write the substitution step before simplifying. That makes sign mistakes easier to catch.