Two real roots
x^2 - 3x + 2 = 0x = 2, 1
Quadratic Formula Calculator
Use this free quadratic formula calculator to solve ax^2 + bx + c = 0, read the discriminant, find real or complex roots, check vertex and axis details, copy steps, and review recent answers.
Real and complex rootsDiscriminantVertex and axisStep-by-step work
Standard formx^2 - 3x + 2 = 0
The discriminant is positive, so the equation has two real roots.x = 2, 1
- Discriminant
- 1
- Vertex
- (1.5, -0.25)
- Axis
- x = 1.5
Steps
- Start with standard form: x^2 - 3x + 2 = 0.
- Use a = 1, b = -3, and c = 2.
- Find the discriminant: b^2 - 4ac = (-3)^2 - 4(1)(2) = 1.
- Use the quadratic formula: x = (-b +/- sqrt(discriminant)) / 2a, with 2a = 2.
- The roots are x = 2, 1.
Graph details
Opensup
Y-intercept2
Root typetwo real
Recent answers
Recent quadratic answers will appear here.
Input tips
Enter the equation in standard form ax^2 + bx + c = 0.
Coefficient a cannot be zero. If a is zero, the equation is linear instead of quadratic.
How to use the Quadratic Formula Calculator
- Enter coefficients a, b, and c from standard form ax^2 + bx + c = 0.
- Press Calculate roots to use the quadratic formula.
- Review the roots, discriminant, vertex, axis of symmetry, and step-by-step work.
- Use examples, recent answers, or copy the answer while you study or graph.
What people use it for
Solve quadratic equations after you identify a, b, and c from standard form ax^2 + bx + c = 0.
Check whether the discriminant gives two real roots, one repeated real root, or complex roots.
Work through equations with negative coefficients, zero b or c terms, and decimal coefficients.
Find graph details such as the vertex, axis of symmetry, opening direction, and y-intercept.
Compare formula roots with a factoring answer or a graphing sketch before you turn in work.
Copy roots and formula steps while keeping recent answers private in the current browser tab.
Quick examples
Negative constant
2x^2 + 5x - 3 = 0x = 0.5, -3
Repeated root
x^2 - 4x + 4 = 0x = 2
Complex roots
x^2 + 2x + 5 = 0x = -1 +/- 2i
Opens downward
-16x^2 + 64x = 0x = 0, 4
Decimal coefficients
0.5x^2 - 3x + 4 = 0x = 4, 2
Need the guide or a nearby tool?
Need a slower walkthrough, a related calculator, or the full library? These links keep you close to the task you started.
- How to use the Quadratic Formula CalculatorSee slower examples, input meanings, formula notes, result limits, and mistakes to avoid.
- CalculatorsBrowse related calculators and utilities in the same category.
- Calculator and utility guidesFind more guides that use real tool inputs and example numbers.
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Frequently asked questions
Quick answers about the quadratic formula, discriminants, complex roots, graph details, coefficients, and privacy.
What formula does the Quadratic Formula Calculator use?
It uses x = (-b +/- sqrt(b^2 - 4ac)) / 2a for equations written in standard form ax^2 + bx + c = 0.
What is the discriminant?
The discriminant is b^2 - 4ac. It tells you the root type: positive means two real roots, zero means one repeated real root, and negative means two complex conjugate roots.
What do the main Quadratic Formula Calculator inputs mean?
The main inputs are the numbers, operation, mode, or known values the calculator needs. Keep units consistent, enter percentages the way the page label shows, and use the examples as a quick check before trusting the answer.
How should I read the Quadratic Formula Calculator answer?
Read the headline answer, then check the supporting lines and examples to understand how the calculator got there. If one input changes, rerun the tool and compare the new answer instead of guessing.
What do a, b, and c mean?
They are the coefficients from standard form ax^2 + bx + c = 0. The a value is attached to x^2, b is attached to x, and c is the constant term.
Why can coefficient a not be zero?
A quadratic equation needs an x^2 term. If a is zero, the equation becomes linear, so the quadratic formula does not apply.
Can this calculator show complex roots?
Yes. When the discriminant is negative, the calculator shows the complex conjugate roots using i.
Does this calculator show graph details?
Yes. It shows the vertex, axis of symmetry, y-intercept, and whether the parabola opens up or down.
What if b or c is zero?
That is fine. Enter 0 for the missing x term or constant term, as long as a is not zero.
Can I enter decimals or negative coefficients?
Yes. You can enter positive, negative, and decimal coefficients such as a = 0.5, b = -3, and c = 4.
What form should I enter the equation in?
Enter the coefficients from standard form ax^2 + bx + c = 0. For example, x^2 - 3x + 2 = 0 uses a = 1, b = -3, and c = 2.
How can I check a root from the answer?
Substitute the root back into ax^2 + bx + c. A correct real root should make the expression equal 0, apart from small rounding differences.
What if my equation is factored?
Expand it into standard form before using this calculator, or use the factor form to read the roots directly if each factor is already simple.
Should I use this instead of a graphing calculator?
Use this for exact formula steps and root type. Use a graphing tool when you need a full visual curve, scale, intercept picture, or context from an applied problem.
What mistakes should I check before trusting the roots?
Check that the equation is in standard form, the signs on b and c are correct, a is not zero, and any decimal coefficients were copied accurately.
Is my quadratic calculation history private?
Yes. Recent quadratic answers stay only in the current browser tab while you use the page. They are not sent to a server.