Quick start
- Rewrite the equation so one side is zero: ax^2 + bx + c = 0.
- Enter a from the x^2 term. It cannot be 0.
- Enter b from the x term, including the sign.
- Enter c from the constant term, including the sign.
- Press Calculate roots.
- Read the roots, discriminant, graph details, and formula steps before copying the answer.
Best uses
The Quadratic Formula Calculator is most useful when factoring is slow, the roots are decimals, the discriminant might be negative, or you need formula steps to show your work.
- 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.
Quick answer
For x^2 - 3x + 2 = 0, enter a = 1, b = -3, and c = 2. The calculator uses x = (-b +/- sqrt(b^2 - 4ac)) / 2a and gives x = 2 and x = 1.
Those roots mean the quadratic equals zero when x is 2 or 1. If you plug either value back into x^2 - 3x + 2, the result is 0.
What each coefficient means
The coefficient a is the number attached to x^2. The coefficient b is the number attached to x. The constant c is the number without x.
Signs matter. In x^2 - 3x + 2 = 0, b is -3, not 3. In 2x^2 + 5x - 3 = 0, c is -3, not 3.
- If there is no written number before x^2, a is 1.
- If the equation starts with -x^2, a is -1.
- If a term is missing, enter 0 for that coefficient.
Rewrite the equation first
The quadratic formula expects everything on one side and 0 on the other. If your problem says x^2 = 3x - 2, move the right side over first so it becomes x^2 - 3x + 2 = 0.
This step is where many wrong answers start. Moving a term across the equals sign changes its sign, so 3x becomes -3x and -2 becomes +2.
- x^2 = 3x - 2 becomes x^2 - 3x + 2 = 0.
- 2x^2 + 5x = 3 becomes 2x^2 + 5x - 3 = 0.
- x^2 + 2x = -5 becomes x^2 + 2x + 5 = 0.
Read the discriminant
The discriminant is b^2 - 4ac. It tells you what kind of roots the equation has before you even read the final root line.
A positive discriminant gives two real roots. A zero discriminant gives one repeated real root. A negative discriminant gives two complex roots with i.
- For x^2 - 3x + 2 = 0, the discriminant is 1, so there are two real roots.
- For x^2 - 4x + 4 = 0, the discriminant is 0, so x = 2 is repeated.
- For x^2 + 2x + 5 = 0, the discriminant is -16, so the roots are complex.
Complex roots are not an error
If the discriminant is negative, the square root part uses i, the imaginary unit. That does not mean the calculator failed. It means the parabola does not cross the x-axis.
For x^2 + 2x + 5 = 0, the calculator reports x = -1 +/- 2i. The matching graph detail still gives the vertex and axis, but there are no real x-intercepts.
Use the graph details
The roots tell you where the graph crosses the x-axis when real roots exist. The vertex tells you where the parabola reaches its highest or lowest point.
The axis of symmetry is x = -b / 2a. If a is positive, the parabola opens up. If a is negative, it opens down. The constant c is the y-intercept.
- Use roots when you need the zero points.
- Use the vertex when you need the maximum or minimum point.
- Use the axis of symmetry when you are sketching or checking a graph.
Check the roots before trusting them
A calculator can solve the equation you entered, but it cannot know whether you copied the original problem correctly. The fastest check is to plug each root back into the equation.
For decimal roots, tiny rounding differences are normal. A result like 0.0000001 is usually rounding noise, not a new answer, as long as the exact steps match the problem.
- Check that a is not 0.
- Check every sign after moving terms to one side.
- Check one root by substitution before turning in the answer.
Common mistakes
The biggest mistake is entering b or c with the wrong sign. The second is using the formula before the equation is equal to 0.
Another mistake is treating a missing term as blank instead of 0. If the equation is x^2 - 9 = 0, b is 0 and c is -9.
- Do not enter 3 for b when the equation has -3x.
- Do not use the quadratic formula for a linear equation where a = 0.
- Do not assume complex roots are wrong just because they include i.
When not to rely on it alone
The calculator is a math helper, not a full algebra tutor. It shows the formula path and graph details, but you still need to choose the right equation and interpret the result for your class, model, or graphing task.
If your problem is really about simplifying radicals, checking powers, or graphing a larger equation, use the nearby tools below instead of forcing everything through the quadratic formula.
Worked examples for Quadratic Formula Calculator
x = 2, 1
x = 0.5, -3
x = 2
x = -1 +/- 2i
x = 0, 4
x = 4, 2
FAQ in plain language
What form does the calculator need?
Use standard form: ax^2 + bx + c = 0. Move all terms to one side before entering a, b, and c.
What if there is no x term?
Enter 0 for b. For example, x^2 - 9 = 0 has a = 1, b = 0, and c = -9.
What if there is no constant term?
Enter 0 for c. For example, x^2 - 4x = 0 has a = 1, b = -4, and c = 0.
Why does a have to be nonzero?
If a is 0, the x^2 term disappears and the equation is linear, not quadratic. The quadratic formula no longer applies.
What does a negative discriminant mean?
It means the roots are complex. The graph does not cross the x-axis, but the equation still has two complex conjugate roots.
How do I check a root?
Substitute the root into the original equation. If the left side equals 0, or is very close because of rounding, the root checks out.
Can I use decimals and negative coefficients?
Yes. Decimals and negative coefficients work as long as you enter the signs correctly and keep the equation in standard form.
Sources
Use these if you want to compare the formula, inputs, or limits with a trusted outside explanation.
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Keep exploring
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