Math Tools
Quadratic Formula Calculator
Solve quadratic equations of the form ax² + bx + c = 0. Enter a, b, and c to find the discriminant, real roots, repeated roots, or complex roots.
Quadratic Formula Calculator
Solve quadratic equations and find real or complex roots instantly. Enter the values of a, b, and c from ax² + bx + c = 0.
Equation
1x² + -3x + 2 = 0
Result
Discriminant
1
x₁
2
x₂
1
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How it works
This calculator solves equations in the form ax² + bx + c = 0 using the quadratic formula. It can return two real roots, one repeated real root, or two complex roots depending on the discriminant.
What is a quadratic formula calculator?
A quadratic formula calculator is a tool that solves quadratic equations in standard form, written as ax² + bx + c = 0. By entering the values of a, b, and c, you can instantly find the roots of the equation without working through each algebra step by hand.
This is useful for students, teachers, exam preparation, homework, algebra practice, and checking answers quickly. It is also helpful when an equation has two real roots, one repeated root, or complex roots.
Quadratic formula explained
The quadratic formula is used to solve any quadratic equation in the form ax² + bx + c = 0, where a is not zero.
Formula
x = (-b ± √(b² - 4ac)) / 2a
In this formula, a, b, and c are the coefficients from the quadratic equation. The value inside the square root, b² - 4ac, is called the discriminant and determines the type of roots the equation has.
Discriminant guide for quadratic equations
The discriminant tells you how many roots the equation has and whether they are real or complex.
| Discriminant value | Root type | Meaning |
|---|---|---|
| b² - 4ac > 0 | Two real roots | The quadratic equation has two different real solutions. |
| b² - 4ac = 0 | One repeated real root | The quadratic touches the x-axis at one point. |
| b² - 4ac < 0 | Two complex roots | The quadratic has no real x-intercepts and gives complex solutions. |
Worked examples using the quadratic formula
These examples show how different coefficient values change the discriminant and the roots.
| Equation | a, b, c | Discriminant | Root type | Roots |
|---|---|---|---|---|
| x² - 3x + 2 = 0 | 1, -3, 2 | 1 | Two real roots | x = 1, 2 |
| x² - 4x + 4 = 0 | 1, -4, 4 | 0 | One repeated real root | x = 2 |
| x² + 2x + 5 = 0 | 1, 2, 5 | -16 | Two complex roots | x = -1 ± 2i |
| 2x² + 5x - 3 = 0 | 2, 5, -3 | 49 | Two real roots | x = 0.5, -3 |
How to solve a quadratic equation step by step
First, write the equation in standard form: ax² + bx + c = 0. Then identify the values of a, b, and c.
Next, calculate the discriminant using b² - 4ac. This tells you whether the equation has two real roots, one repeated real root, or two complex roots.
Finally, substitute the values into the quadratic formula and simplify. This calculator performs those steps instantly and shows the root type along with the result.
When to use the quadratic formula
The quadratic formula is especially useful when a quadratic equation cannot be solved easily by factoring. It works for all quadratic equations as long as a is not zero.
Students often use it in algebra, pre-calculus, and exam prep. Teachers use it to demonstrate root behavior, and professionals may use quadratic equations in physics, engineering, optimization, and graph analysis.
Common mistakes when using the quadratic formula
One common mistake is entering the wrong signs for b or c. For example, x² - 3x + 2 = 0 means b is -3, not 3. Sign errors can completely change the discriminant and the final roots.
Another mistake is forgetting that a cannot be zero. If a is zero, the equation is not quadratic. It becomes a linear equation instead and should be solved using a different method.
Users also sometimes forget to handle negative discriminants as complex roots. When b² - 4ac is negative, the square root involves the imaginary unit i.
Quadratic formula calculator FAQ and common algebra questions
What is the quadratic formula?
The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. It is used to solve quadratic equations written in the form ax² + bx + c = 0.
How do you solve a quadratic equation?
Identify a, b, and c, calculate the discriminant b² - 4ac, and substitute the values into the quadratic formula to find the roots.
What does the discriminant tell you?
The discriminant tells you the type of roots. Positive means two real roots, zero means one repeated real root, and negative means two complex roots.
Can a quadratic equation have complex roots?
Yes. If the discriminant is negative, the equation has two complex roots instead of real roots.
What is the standard form of a quadratic equation?
The standard form is ax² + bx + c = 0, where a, b, and c are constants and a is not zero.
What happens if a is zero?
If a is zero, the equation is not quadratic. The quadratic formula should not be used because the equation becomes linear.
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