This quadratic formula calculator is provided by Hesapstan to solve ax²+bx+c=0 and show the discriminant, real or complex roots, exact forms where suitable, and decimal approximations.
This calculator solves a quadratic equation with the quadratic formula
The Quadratic Formula Calculator solves equations in the form ax²+bx+c=0. It takes the coefficients a, b, and c, then shows the formula, the substitution step, the discriminant, and the two roots.
- It computes the discriminant Δ.
- It classifies the roots as two real roots, one repeated real root, or two complex roots.
- It shows exact rational or simplified radical forms where reasonable.
- It always includes decimal approximations for checking.
If a=0, the expression is not a quadratic equation. The calculator rejects that case instead of silently dividing by zero or returning a misleading quadratic result.
What is the quadratic formula?
The quadratic formula finds the roots of ax²+bx+c=0 using x=(-b±√Δ)/(2a), where Δ=b²−4ac. The value of Δ decides whether the roots are real, repeated, or complex.
The formula is useful when factoring is not obvious. For example, x²−2x−1=0 does not factor nicely over integers, but the formula gives x=1±√2 directly.
This calculator applies the formula directly. Completing the square can reach the same roots through a different learning path, but this page focuses on substitution, discriminant, and root values.
How does the discriminant classify the roots?
The discriminant Δ=b²−4ac determines the type of roots a quadratic equation has. The calculator computes Δ first, then displays the roots according to that case.
- If Δ>0, there are two distinct real roots.
- If Δ=0, both roots collapse into one repeated root.
- If Δ<0, the equation has no real roots but has a complex conjugate pair.
- If Δ is a perfect square, the exact roots are often integers or fractions.
- If Δ is not a perfect square, a simplified radical and a decimal approximation may be shown.
This is more than a discriminant-only check: after showing Δ, the calculator returns the actual root values.
How the calculator builds the result
The calculator substitutes a, b, and c into the quadratic formula, computes Δ, then chooses the clearest root format. Root cards indicate whether a value is exact or approximate.
- Read the coefficients a, b, and c.
- Compute Δ=b²−4ac.
- Classify the root type from the sign of Δ.
- Compute the roots from x=(-b±√Δ)/(2a).
- Show a fraction or simplified radical when suitable.
- Show decimal approximations for comparison.
The ≈ label means the decimal value is rounded. It is useful for checking, but if an exact form is also shown, the exact form is the mathematical answer.
Example: two distinct real roots
For x²−5x+6=0, the coefficients are a=1, b=−5, and c=6. The discriminant is Δ=25−24=1, so the equation has two distinct real roots.
- In the formula, −b=5 and 2a=2.
- √Δ=√1=1.
- x₁=(5+1)/2=3.
- x₂=(5−1)/2=2.
Both roots can be displayed exactly: 3 and 2. The decimal approximations simply confirm the same values.
Example: a simplified radical result
For x²−2x−1=0, the coefficients are a=1, b=−2, and c=−1. The discriminant is Δ=4+4=8, which is positive but not a perfect square.
- The formula gives x=(2±√8)/2.
- √8 simplifies to 2√2.
- The exact roots are x=1±√2.
- The approximate roots are x≈2.414 and x≈−0.414.
When the radical can be simplified, the calculator avoids pretending that the decimal is exact. The decimal value is a practical approximation.
How to read complex roots
When Δ is negative, the roots are not real numbers; they appear as a complex conjugate pair. For x²+2x+5=0, Δ=4−20=−16, so the roots are −1+2i and −1−2i.
This is not an error. It means the parabola does not cross the x-axis, while the equation still has two roots in the complex number system.
For Δ<0, the equation has no real roots, but it does have complex roots. The calculator labels them as complex so the distinction is visible.
How this differs from a discriminant calculator
This calculator solves for the roots. A discriminant calculator only computes Δ and explains the root type. If you need the actual x-values, this quadratic formula calculator is the right tool.
- Discriminant calculator: What is Δ and what kind of roots exist?
- Quadratic formula calculator: What are the actual x roots?
- Completing the square: Solve the same equation through a different method.
- Graphing quadratic inequalities: Use roots as boundary points for inequality intervals.
Common mistakes
Most quadratic formula mistakes come from signs, the discriminant, or confusing exact and approximate results.
- Writing −b with the wrong sign when b is negative.
- Forgetting that b² is always the square of the whole b value.
- Dropping the sign of 4ac.
- Trying to use the formula when a=0.
- Treating an approximate decimal as an exact answer.
- Thinking Δ<0 is a calculator error instead of a complex-root case.
Limitations of this calculator
This calculator is limited to quadratic equations in one variable. It does not draw a graph, solve inequalities, or solve cubic equations.
If you need a parabola sketch or an inequality solution set, use the graphing or inequality calculators instead. This page focuses on root solving.
The formula-based result is mathematically stable, but decimal approximations are rounded. Use exact forms when they are available.
Frequently Asked Questions
What equations can the quadratic formula solve?
It solves equations in the form ax²+bx+c=0 where a is not zero. If a=0, the equation is not quadratic.
Does this calculator only compute the discriminant?
No. It computes Δ, explains the root type, and returns the actual x roots.
Is a negative discriminant an error?
No. A negative discriminant means there are no real roots, but there are two complex conjugate roots.
What is the difference between exact and approximate roots?
An exact root is a fraction or radical expression. An approximate root is a rounded decimal value marked with ≈.
Why does this calculator not include a graph?
This page focuses on solving the roots. Graphing and inequality behavior belong to separate graphing or inequality calculators.