Quadratic Formula Calculator and Quadratic Equation Solver
Our quadratic formula calculator solves any equation of the form ax² + bx + c = 0 and shows every step. It works as a quadratic equation calculator for real and complex roots, with the vertex, factored form and a graph. Type a, b and c to get your answer.
What Is a Quadratic Equation
A quadratic equation has x² as its highest power and can be written as ax² + bx + c = 0, where a is not zero. Its graph is a U shaped curve called a parabola. The solutions, or roots, are the points where that curve crosses the x axis.
How to Use the Quadratic Formula Calculator
- Move every term to one side so the other side is zero.
- Enter a, b and c, including any minus signs.
- Read both roots, the exact form and the steps.
- Check the graph, where green dots mark the real roots and a red dot marks the vertex.
If a is 0, the equation is linear and the quadratic equation calculator solves bx + c = 0 instead. Decimal coefficients work too, but the exact radical form only appears for whole numbers. The quadratic equation calculator updates on every keystroke, so you can test changes quickly.
The Quadratic Formula
The formula is x = (−b ± √(b² − 4ac)) / 2a. The ± sign gives two answers, one with a plus and one with a minus. Every quadratic can be solved this way, even when it will not factor neatly.
Take x² − 5x + 6 = 0, where a = 1, b = −5 and c = 6. The discriminant is 25 − 24 = 1, so x = (5 ± 1) / 2. That gives roots of 3 and 2, which the quadratic formula calculator shows with every line of working.
What the Discriminant Tells You
The discriminant, D = b² − 4ac, reveals the type of roots before you finish solving. It is the part under the square root in the formula. Its sign decides how many times the graph meets the x axis.
| Discriminant | Roots | Graph |
| D > 0 | Two different real roots | Crosses the x axis twice |
| D = 0 | One repeated real root | Touches the x axis once |
| D < 0 | Two complex roots | Never meets the x axis |
For x² + 4x + 4 = 0, D is 0 and the only root is −2. For x² + 2x + 5 = 0, D is −16, so the roots are −1 + 2i and −1 − 2i. Our quadratic equation calculator labels each case for you.
Exact Answers and Radicals
Decimal answers are handy, but homework often asks for exact form. For x² − 4x + 1 = 0, D is 12 and √12 simplifies to 2√3. The quadratic formula calculator reduces the answer to x = 2 ± √3, which is about 3.7321 and 0.2679.
When D is a perfect square, the roots are whole numbers or fractions. For 2x² + 3x − 2 = 0, D is 25 and the roots are 1/2 and −2. A good quadratic calculator should show these as clean fractions, not long decimals.
Other Ways to Solve a Quadratic
Factoring
If you can find two numbers that multiply to ac and add to b, the equation factors. For x² − 5x + 6, those numbers are −2 and −3, so it becomes (x − 2)(x − 3) = 0. Factoring is fast, but many equations do not have neat factors.
Completing the Square
Completing the square rewrites the equation as a(x − h)² + k = 0. It is how the quadratic formula itself is derived, and it gives the vertex directly. Many teachers ask for it when the lesson covers graphing.
Vertex of a Parabola
The vertex is the highest or lowest point of the curve. Its x value is h = −b / 2a, and you find k by putting h back into the equation. For x² − 5x + 6, the quadratic equation calculator puts the vertex at (2.5, −0.25) and the axis of symmetry is x = 2.5.
When a is positive the parabola opens upward and the vertex is a minimum. When a is negative it opens downward and the vertex is a maximum. Finding the vertex of a parabola answers many largest or smallest questions in word problems.
A Real World Example
A ball is thrown upward from 1.5 meters at 20 meters per second. Its height after t seconds is roughly h = −4.9t² + 20t + 1.5. Set h to 0 and enter a = −4.9, b = 20 and c = 1.5 in the quadratic equation calculator to get roots of 4.1553 and −0.0737.
Only the positive root makes sense, so the ball lands after about 4.16 seconds. The vertex at (2.0408, 21.9082) shows it reaches a peak of about 21.91 meters after 2.04 seconds. Drop negative times and other impossible answers in word problems.
Common Mistakes
- Dropping the minus sign when b is negative, since −b then becomes positive.
- Squaring a negative b wrongly, because (−5)² is 25, not −25.
Why Use Our Quadratic Formula Calculator
It gives decimal and exact roots, the discriminant, vertex, factored form and a graph in one place. Everything runs in your browser and nothing is saved. Enter a, b and c and solve your equation now.
Frequently asked questions
Is this quadratic formula calculator free?
Yes, it is free with no sign up and no limit.
Can the quadratic equation calculator find complex roots?
Yes, it shows them in a + bi form when the discriminant is negative.
Is this a quadratic equation solver for decimals?
Yes, decimals work, but exact radical form needs whole numbers.
Why does a quadratic calculator give two answers?
The ± in the formula produces one root with a plus and one with a minus.
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