Solve Ax2 Bx C Calculator
Use this solve ax2 bx c calculator to solve ax^2 + bx + c = 0 in a beginner-friendly way. Enter coefficient a, coefficient b, and coefficient c to calculate the discriminant, classify the roots, and follow a clean step-by-step breakdown that shows exactly how the quadratic formula is applied.
All calculations use standard published formulas. Results are for informational use only.
x^2 - 5x + 6 = 0 has two distinct real roots because D = 1 is positive. The solutions are x1 = 3 and x2 = 2.
Discriminant summary
D = 1. The discriminant is positive, so the equation has two different real roots.
| Discriminant | Meaning | What it means for the graph |
|---|---|---|
| D > 0 | Two distinct real roots | The parabola crosses the x-axis twice. |
| D = 0 | One repeated real root | The parabola touches the x-axis once at the vertex. |
| D < 0 | Two complex roots | The parabola does not cross the x-axis in the real plane. |
Formula used
The calculator uses the standard-form quadratic workflow: write the equation as ax^2 + bx + c = 0, compute D = b^2 - 4ac, then apply x = (-b +/- sqrt(D)) / (2a).
| Part | Expression | Why it matters |
|---|---|---|
| Standard form | ax^2 + bx + c = 0 | The coefficients a, b, and c come from this exact arrangement. |
| Discriminant | D = b^2 - 4ac | Use D to predict whether the roots are real, repeated, or complex. |
| Quadratic formula | x = (-b +/- sqrt(D)) / (2a) | This is the universal method for solving quadratic equations. |
| Real root test | D > 0, D = 0, or D < 0 | The sign of D controls the type of root the equation has. |
Substitution breakdown
Step-by-step solution
Quadratic equations are solved most cleanly when they are written as ax^2 + bx + c = 0.
These three coefficients plug directly into the discriminant and the quadratic formula.
The discriminant tells you what kind of roots to expect before you finish the full calculation.
Square b first, then subtract 4ac.
The discriminant is positive, so the equation has two different real roots.
The same formula works for every quadratic equation as long as a is not 0.
Replace a, b, and D with the numbers from your equation.
Take the + branch once and the - branch once.
These are the two x-values that make the equation equal to 0.
Interpretation
The discriminant is positive, so the equation has two different real roots.
Because D = 1 is greater than 0, sqrt(D) is a real non-zero number. That makes the + and - branches of the quadratic formula land on two different real answers. On a graph, the parabola crosses the x-axis at two points.
Introduction
This solve ax2 bx c calculator is built for people who want more than a bare answer. If you are searching for a way to solve ax^2 + bx + c = 0, you usually need the roots, the discriminant, the root classification, and a readable explanation of what happened. That is exactly what this page delivers. You can enter coefficients, check the final roots, and then read the working line by line in plain notation that stays friendly on desktop and mobile.
The page is especially useful for homework checks, exam revision, and quick algebra verification. Instead of treating the quadratic formula like a black box, it shows how the standard form equation connects to D = b^2 - 4ac and then to x = (-b +/- sqrt(D)) / (2a). That makes the result easier to trust and much easier to learn from.
What ax^2 + bx + c = 0 means
The expression ax^2 + bx + c = 0 is called the standard form of a quadratic equation. The coefficient a belongs to x^2, b belongs to x, and c is the constant term. The reason the equation must be arranged this way is simple: those positions are exactly the positions used by the quadratic formula. When the order is clear, the substitution step becomes mechanical and far less error-prone.
A quadratic behaves differently from a linear equation because the highest power of x is 2. Graphically, that means the equation produces a parabola rather than a straight line. Depending on the coefficients, that parabola can cross the x-axis twice, touch it once, or miss it completely in the real number system. Those three possibilities are exactly what the discriminant measures.
How to solve a quadratic equation in standard form
The most reliable workflow is to begin in standard form quadratic solver, identify a, b, and c, compute the discriminant, and then apply the quadratic formula. This avoids guessing and works even when the roots do not factor neatly. It also helps you predict the type of answer before you do the last arithmetic step. That matters because a negative discriminant should immediately warn you that the final roots will involve i rather than ordinary real numbers.
This page is designed to show quadratic formula with steps, not just the result. The step generator writes the equation, identifies the coefficients, calculates D = b^2 - 4ac, interprets the sign of D, substitutes into x = (-b +/- sqrt(D)) / (2a), and then simplifies to the final roots. If you are learning the method, that structure mirrors the way a teacher would explain it on paper.
Discriminant explained
The discriminant is the quantity inside the square root once the quadratic formula is prepared. It is written as D = b^2 - 4ac. This one value does a surprising amount of work. A positive discriminant means two different real roots. A zero discriminant means one repeated real root. A negative discriminant means two complex roots. That is why a good discriminant calculator is often the fastest way to understand the entire equation.
The sign of D also tells you something visual. If D > 0, the parabola crosses the x-axis twice. If D = 0, it touches the x-axis once at the vertex. If D < 0, it stays above or below the x-axis and never crosses it in the real plane. That connection between algebra and graph behavior is one of the most important ideas in beginning quadratic equations.
Quadratic formula explained
The quadratic formula is x = (-b +/- sqrt(D)) / (2a). It is universal, which means it works for every genuine quadratic equation. That makes it more dependable than factoring, even if factoring is sometimes faster when the numbers are simple. A solid quadratic formula with steps should therefore do three things well: keep the notation readable, show both branches clearly, and explain where the discriminant fits into the calculation.
The plus branch gives one root and the minus branch gives the other. If D is zero, those two branches collapse into the same value. If D is negative, the square root creates an imaginary part and the answer becomes a complex conjugate pair. That is why the formula is so powerful: it handles two real roots, a repeated root, and complex roots inside one consistent structure.
How to use this calculator
Enter the coefficient of x^2 into the a field, the coefficient of x into the b field, and the constant into the c field. Then press Calculate. The page returns the final roots, the discriminant, the root type, the substitution breakdown, and a full step-by-step solution. The Reset button clears the form, while the Example button restores a ready-made equation so you can test the workflow instantly.
If you are checking schoolwork, start by comparing the equation preview with your original problem. That simple habit prevents many input mistakes. Then read the discriminant summary before jumping to the roots. It tells you what kind of answer to expect and gives you a quick way to catch sign errors before they spread through the formula.
Common mistakes
The most common input mistake is forgetting to move every term to one side before solving. The quadratic formula assumes the equation already equals 0. Another frequent issue is mixing up the coefficients, especially copying b into a or forgetting a negative sign. Because both the discriminant and the denominator depend on those numbers, even a small sign error can change the whole answer.
Another common confusion is expecting every quadratic to have two ordinary real roots. Some equations have a repeated root calculator, and some need the format used by a quadratic equation with complex roots. That is not a sign that the calculation went wrong. It is simply the correct mathematical outcome for that equation. Reading the discriminant first is the easiest way to avoid that misunderstanding.
Worked examples
These three worked examples show the three main outcomes you need to recognize when solving ax^2 + bx + c = 0: two real roots, one repeated real root, and two complex roots.
Example 1 - Two real roots
Equation: x^2 - 5x + 6 = 0
Values: a = 1, b = -5, c = 6
Discriminant: D = 1
Root type: Two distinct real roots
Final roots: x1 = 3, x2 = 2
Here the discriminant is positive, so the parabola crosses the x-axis twice and the formula produces two different real answers.
Example 2 - One repeated real root
Equation: x^2 - 6x + 9 = 0
Values: a = 1, b = -6, c = 9
Discriminant: D = 0
Root type: One repeated real root
Final roots: x1 = 3, x2 = 3
Here the discriminant is 0, so both branches of the formula collapse to the same answer and the parabola only touches the x-axis once.
Example 3 - Complex roots
Equation: x^2 + 4x + 13 = 0
Values: a = 1, b = 4, c = 13
Discriminant: D = -36
Root type: Two complex roots
Final roots: x1 = -2 + 3i, x2 = -2 - 3i
Here the discriminant is negative, so the roots are complex. The equation is still solved correctly, but the answers live in the complex number system.
How to use this calculator
- Enter the coefficient of x^2 in a, the coefficient of x in b, and the constant in c.
- Press Calculate to solve the equation and classify the roots.
- Check the discriminant summary first so you know whether to expect real, repeated, or complex roots.
- Open the substitution and step-by-step sections to verify each line of the algebra.
- Use Example any time you want a ready-made equation to practice with.
Common mistakes
Forgetting to set the equation equal to 0
The quadratic formula assumes the equation is already written as ax^2 + bx + c = 0. If the expression is not equal to 0 first, the coefficients will be wrong and the roots will be wrong too.
Using the wrong value for a
The value of a is the coefficient of x^2, not the coefficient of x. If a is copied incorrectly, both the denominator 2a and the discriminant term 4ac change, so the whole solution shifts.
Misreading a negative discriminant
A negative discriminant does not mean there is no solution. It means there are no real roots. The correct response is to write the answer as a complex pair with i, not to stop the calculation early.
Dropping one branch of +/-
The quadratic formula usually creates two answers because of the + and - branches. Unless D = 0, you must evaluate both branches to get the complete solution set.
Frequently Asked Questions
How do you solve ax^2 + bx + c = 0?
Write the equation in standard form, identify a, b, and c, calculate the discriminant D = b^2 - 4ac, and then apply x = (-b +/- sqrt(D)) / (2a). This solve ax2 bx c calculator performs those steps automatically and also shows the full substitution so you can follow the algebra instead of only reading the final root values.
What if a is 0?
If a = 0, the equation is no longer quadratic because the x^2 term disappears. In that case the quadratic formula does not apply. The expression becomes a linear equation, so you should solve it with linear methods instead.
What does the discriminant calculator tell you?
The discriminant tells you what kind of roots to expect before you finish the calculation. If D > 0, there are two distinct real roots. If D = 0, there is one repeated real root. If D < 0, the equation has two complex roots.
Can this quadratic equation with complex roots show complex roots?
Yes. When the discriminant is negative, the calculator rewrites the answer using the real part and the imaginary part so you can see the conjugate pair clearly. That makes it useful for algebra classes where teachers expect the final answer in a + bi form.
Why must the equation be in standard form quadratic solver?
The coefficients only match the formula correctly when the equation is arranged as ax^2 + bx + c = 0. If the expression starts in factored form, vertex form, or has terms on both sides, rewrite it first so a, b, and c are unambiguous.
When is a quadratic formula with steps better than factoring?
Factoring is fast when the numbers are simple, but it does not work conveniently for every quadratic. The quadratic formula works every time, including when the roots are irrational, repeated, or complex. That is why formula-based solving is the safer universal method.