Quadratic Formula Calculator
Enter a, b and c from ax² + bx + c = 0. The calculator works the formula through term by term, keeps the answer exact where an exact form exists, and says what the discriminant means before it solves.
How it works
How the Quadratic Formula Calculator works
the part a result on its own leaves out
What the formula does
The quadratic formula gives both roots of any quadratic equation from its three coefficients, whether or not the expression factors. It is the general solution of ax² + bx + c = 0, and it comes from completing the square on that equation once, in general, so nobody has to do it again.
Why the discriminant comes first
b² − 4ac sits under the root and decides the shape of the answer before any arithmetic is done. Positive means two real roots; zero means one repeated root; negative means none, because no real number squares to a negative. This calculator evaluates it as a separate step for that reason.
Common mistakes
Common mistakes with the quadratic formula
the errors that actually show up, not every possible slip
Dividing only part of the numerator by 2a
The whole of −b ± √D sits over 2a, not just the root. Writing −b ± √D/2a changes the answer and is the most common slip in this formula.
Losing the sign of b
The formula starts with −b. If b is already negative, −b is positive: for b = −3 the numerator begins with +3, not −3.
Rounding before the last line
√5 is exact; 2.236 is not. Round once, at the end, or the error compounds through both roots.
Questions
Asked often enough to answer here
answers that change how you work
When should I use the quadratic formula instead of factoring?
Factor first if the numbers look friendly — it is faster. If no integer pair works, or the discriminant is not a perfect square, use the formula. It always works, which factoring does not.
What does a negative discriminant mean?
There are no real roots, and on a graph the parabola never crosses the x-axis. The roots exist as a complex conjugate pair, which the calculator shows.
Where does the quadratic formula come from?
From completing the square on ax² + bx + c = 0 with letters instead of numbers. Doing it once in general produces the formula, so the work does not have to be repeated for every equation.
What do the sum and product of the roots tell me?
They give a fast check. The two roots must add to −b/a and multiply to c/a, so if either fails, the arithmetic went wrong somewhere.
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