Quadratic Formula Calculator
Solve any quadratic equation for real or complex roots with full step by step work.
Equation Solver
| Value | Result |
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Discriminant Outcomes Reference Table
This reference table shows how the discriminant value determines the type of roots a quadratic equation will have.
| Discriminant Value | Condition | Root Type |
|---|---|---|
| b squared minus 4ac > 0 | Positive | Two distinct real roots |
| b squared minus 4ac = 0 | Zero | One repeated real root |
| b squared minus 4ac < 0 | Negative | Two complex conjugate roots |
Quadratic Formula Explained
Quadratic formula explained simply means using a single equation to find the values of x that satisfy any quadratic equation written in the standard form ax squared plus bx plus c equals 0, where a cannot be zero.
The three numbers a, b, and c are called coefficients. The coefficient a multiplies x squared, b multiplies x, and c is the constant term added at the end.
How to Find the Discriminant
How to find the discriminant involves calculating the value of b squared minus 4ac, which sits underneath the square root symbol in the quadratic formula and reveals what kind of solutions exist before solving further.
How to Solve for Real Roots
How to solve for real roots applies when the discriminant is zero or positive. Substitute a, b, and c into the formula, simplify the square root, and compute both the plus and minus versions of x.
How to Solve for Complex Roots
How to solve for complex roots applies when the discriminant is negative. The square root of a negative number introduces the imaginary unit i, and the two solutions become complex conjugates of each other.
Worked Example of the Quadratic Formula
Worked example of the quadratic formula helps illustrate how the steps fit together using real numbers. Consider the equation x squared minus 8x plus 5 equals 0, where a equals 1, b equals negative 8, and c equals 5.
Step by Step Calculation
Step by step calculation begins by substituting the values into the formula to get x equals 8 plus or minus the square root of 64 minus 20, all divided by 2. This simplifies to x equals 8 plus or minus the square root of 44, divided by 2.
Simplifying the Result
Simplifying the result reduces the square root of 44 to 2 times the square root of 11, which then cancels with the denominator to leave x equals 4 plus or minus the square root of 11, giving two real decimal approximations.
When Equations Produce Complex Roots
When equations produce complex roots, such as 5x squared plus 20x plus 32 equals 0, the discriminant becomes negative, and the final answer includes an imaginary component written with the letter i.
Frequently Asked Questions
The quadratic formula is x equals negative b plus or minus the square root of b squared minus 4ac, all divided by 2a. It is used to solve any equation in the form ax squared plus bx plus c equals 0, where a is not zero.
The discriminant is the expression b squared minus 4ac found under the square root in the quadratic formula. Its value determines whether the equation has one real root, two real roots, or two complex roots.
When the discriminant is negative, the equation has two complex roots that include an imaginary number represented by i. These roots are conjugates of each other, differing only in the sign before the imaginary part.
When the discriminant equals zero, the equation has exactly one real root, sometimes called a repeated or double root, because both solutions from the formula are identical.
No, a cannot be zero in the quadratic formula because dividing by 2a would be undefined. If a equals zero, the equation becomes linear rather than quadratic and should be solved differently.
To check if a solution is correct, substitute the value of x back into the original equation ax squared plus bx plus c. If the result equals zero, the solution is correct.