Quadratic Equation Solver
Solve a x²+b x+c=0 with the quadratic formula. The discriminant detects real, double, and complex roots, and the vertex, axis, and parabola graph are shown.
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Input
Enter the coefficients a, b and c of the quadratic equation a x²+b x+c=0. The two roots are found with the quadratic formula.
The equation is a x²+b x+c=0. The coefficient a must not be zero.
Result
Two distinct real roots
Root x₁
2
Root x₂
1
Discriminant D
1
Axis of symmetry
x = 1.5
Vertex
( 1.5 , -0.25 )
The roots are x = ( -b ± sqrt(D) ) / 2a with discriminant D = b² - 4ac. When D is negative the roots are complex.
How it works
- The quadratic formula x = ( -b ± sqrt(D) ) / 2a is used, with discriminant D = b² - 4ac classifying the roots.
- When D is positive there are two distinct real roots, when D is zero there is a double root, and when D is negative there are conjugate complex roots.
- The parabola vertex is ( -b / 2a , c - b² / 4a ) and the axis of symmetry is x = -b / 2a.
- When the coefficient a is zero it is not a quadratic equation, so it cannot be solved.
Frequently asked questions
- How is a quadratic equation solved?
- With the quadratic formula x = ( -b ± √D ) / 2a, where the discriminant D = b² − 4ac determines the type of roots. If the coefficient a is 0 it is not a quadratic, so it cannot be solved.
- How does the discriminant tell the type of roots?
- When D = b² − 4ac is positive there are two distinct real roots, when it is 0 there is a repeated (double) root, and when it is negative there are two conjugate complex roots.
- Does it also show the vertex and axis of symmetry?
- Yes. The parabola's vertex is ( -b / 2a , c - b² / 4a ) and its axis of symmetry is x = -b / 2a, shown together with the graph.
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