Quadratic Equation Solver
Solve any quadratic equation using the quadratic formula. Get real or complex roots instantly.
About This Calculator
Solve any quadratic equation using the quadratic formula. Get real or complex roots instantly. Use the calculator above for instant results.
Understanding Quadratic Equations
A quadratic equation has the form ax²+bx+c=0 with a≠0. Its graph y=ax²+bx+c is a parabola, and solving the equation means finding the x-coordinates where that parabola meets the x-axis. The quadratic formula works for every quadratic and comes from completing the square.
The discriminant Δ=b²−4ac tells you the root type before you take a square root. If Δ>0, there are two distinct real roots. If Δ=0, both roots coincide at x=−b/(2a). If Δ<0, the roots are a complex-conjugate pair. The axis of symmetry is x=−b/(2a), so for two real roots their average equals that symmetry value.
How to Verify Quadratic Roots
The fastest verification is substitution: insert each reported root r into ar²+br+c and confirm the result is zero, allowing for small rounding error. Vieta's relations provide an even quicker pair check. For roots r₁ and r₂, their sum should be -b/a and their product should be c/a.
The graph gives another interpretation. When Δ>0 the parabola crosses the x-axis twice; when Δ=0 it only touches the axis at its vertex; when Δ<0 it never intersects the real x-axis. The vertex's x-coordinate -b/(2a) lies halfway between two real roots, so it is a useful symmetry check.
Coefficient and Graph Clues
The sign of a controls the opening direction of the parabola: a>0 opens upward and a<0 opens downward. The constant c is the y-intercept because f(0)=c. If c=0, then x=0 is automatically one root and the equation can be factored as x(ax+b)=0. These observations provide fast checks before applying the full formula.
Worked Examples
- Discriminant: 25-24=1 | x₁=3, x₂=2
Answer: x=3 and x=2
- Discriminant: 16-16=0 | One root: x=1
Answer: x=1 (repeated)
- Discriminant: -3 < 0 | Complex roots: no real solutions
Answer: Complex roots
- a=1,b=0,c=-9; Δ=36.
- x=(0±6)/2.
Answer: x=3 and x=-3
- Δ=36−36=0.
- x=-6/6=-1.
Answer: x=-1 (repeated)
Who Uses This Calculator?
Solve quadratic equations quickly.
Find zeros of polynomial systems.
Demonstrate the quadratic formula.
Solve optimization problems.
Common Mistakes to Avoid
❌ Forgetting to check the discriminant
Discriminant = b²-4ac tells you the number of real solutions before solving: >0 = two roots, =0 = one root, <0 = no real roots.
❌ Sign errors with b
In the formula x = (-b ± √disc)/(2a), don't forget the negative sign in front of b.
❌ Forgetting the denominator applies to the entire numerator
Use x=(-b±√Δ)/(2a). Both -b and the square-root term are divided by 2a.
❌ Reporting “no solution” when Δ<0
There are no real roots, but there are still two complex roots when complex numbers are allowed.
Substituting each reported root back into ax²+bx+c should produce zero apart from ordinary rounding error.
Frequently Asked Questions
Quadratic formula?
x = (-b ± √(b²-4ac)) / (2a). Works for any quadratic equation ax²+bx+c=0.
What is the discriminant?
b²-4ac. If >0: two real roots. If =0: one root. If <0: no real roots (complex solutions).
What does 'solving a quadratic' mean?
Finding the value(s) of x where the parabola y=ax²+bx+c crosses the x-axis (where y=0).
Why must a be nonzero?
If a=0, the x² term disappears and the equation becomes linear rather than quadratic.
How are the roots related to the coefficients?
For roots r₁ and r₂, r₁+r₂=-b/a and r₁r₂=c/a. These are Vieta's relations.
Can a quadratic have exactly one real root?
Yes. When Δ=0 the two algebraic roots are equal, giving one distinct real x-value with multiplicity two.
How can I check my roots?
Substitute each root into ax²+bx+c. The result should be zero apart from rounding error.
Formula Explorer connections
Interpretation: the discriminant Δ=b²−4ac is the validity gate: positive gives two real crossings, zero gives one repeated root, and negative moves the roots into the complex plane. Assumption: a≠0; otherwise the equation is linear.