Quadratic Formula Calculator
This quadratic formula calculator solves ax² + bx + c = 0 exactly — and it does two things the others do not: it reduces the surd all the way, and it checks both roots against your own a, b and c before showing them to you.
Type 1, −4, 1 and you get x = 2 ± √3. Not (4 ± √12) ÷ 2, which is the same number and half an answer.
How the Quadratic Formula Works
x = (−b ± √(b² − 4ac)) ÷ 2a
Three pieces, and each does a separate job. The −b ÷ 2a is the middle of the two answers — it is also where the parabola turns. The √(b² − 4ac) is how far either side of that middle you have to go. And the ± says you go both ways, which is why a quadratic has two roots and not one.
Factoring is faster when it works. It works on a small minority of the quadratics you will actually be given, and you cannot tell which in advance. This formula works on all of them, which is the argument for knowing it cold.
The Sign of b, and Why It Costs Marks
More marks are lost here than anywhere else in the topic, and the reason is almost silly: the formula opens with −b, and when b is already negative, −b is positive.
For x² − 4x + 1 = 0, b is −4. So −b is +4, not −4:
x = (−(−4) ± √12) ÷ 2 → x = (4 ± √12) ÷ 2
Write that first line out. Every time. The tidied version looks perfectly reasonable with a −4 in it, which is exactly the problem — there is nothing on the page to tell you it is wrong, and both roots come out with their signs flipped. This calculator prints the untidied substitution above the neat one for the same reason.
What the Discriminant Tells You Before You Solve
Before working out either root, take b² − 4ac on its own. It settles the kind of answer:
| Positive | two real roots | x² − 3x + 2 gives 1 and 2 |
| Zero | one repeated root | x² − 6x + 9 gives 3 |
| Negative | a complex pair | x² + 2x + 5 gives −1 ± 2i |
The middle row is worth pausing on. When the discriminant is zero, both branches of the formula give the same number, so the answer is written once. Handing in the same value twice is a real and common way to lose a mark on a question you had entirely right.
Simplifying the Answer: Getting the Square Out of the Root
Here is where most answers — and most calculators — stop one step early.
(4 ± √12) ÷ 2 is not finished, because 12 has a square hiding in it. Pull it out:
√12 = √(4 × 3) = 2√3
x = (4 ± 2√3) ÷ 2 = 2 ± √3
Once the 2 comes out of the root, the top and bottom share it and the whole fraction cancels. A marker who wanted 2 ± √3 will not accept the earlier line, even though the two are the same number to the last decimal place.
The same applies to complex roots, and it is skipped even more often there. (−2 ± 4i) ÷ 2 is −1 ± 2i. Do the division; it does not stop being a fraction just because there is an i in it.
Writing the Answer the Way It Is Marked
Two answers can be equal and only one of them get full credit. Watch what happens with x² − x − 1 = 0:
(1 ± √5) ÷ 2 and ½ ± ½√5
Identical numbers. The first is what the back of the book says, and the second invites a genuine misreading — is that half of √5, or is the 2 under the root with the 5? Keep the single fraction whenever the denominator survives.
And when it does not survive, do not put it back. 2 ± √3 should not be written as (4 ± 2√3) ÷ 2 for the sake of looking like the formula. This calculator switches between the two forms on exactly that rule: one fraction while there is a denominator, no fraction once it has cancelled.
When the Quadratic Formula Gives Complex Roots
A negative discriminant is not a dead end and it is not a mistake. It means the parabola never reaches the x-axis, so the two roots are a conjugate pair — the same number with the sign of its imaginary part flipped.
Take x² + 2x + 5 = 0. D = 4 − 20 = −16, and √−16 is 4i, so:
x = (−2 ± 4i) ÷ 2 → x = −1 ± 2i
That last step is the part people skip. It is the identical cancellation you would do on a surd — the 2 comes out of the root before the i goes in, and then the top and the bottom share a factor. Complex answers get left unreduced far more often than real ones, purely because they feel unfamiliar enough that nobody thinks to tidy them.
When the root does not come out whole, the fraction stays: x² + x + 1 = 0 gives (−1 ± √3 i) ÷ 2. Same rule as before, applied to the same three parts.
Checking the Roots by Their Sum and Product
Nobody substitutes an irrational root back into the original equation. It is unpleasant, it takes longer than the question did, and so the check quietly never happens.
There is a much easier one, and it is already in your syllabus:
the two roots add to −b ÷ a
the two roots multiply to c ÷ a
Take 2x² + 5x − 3 = 0, whose roots are −3 and ½. Their sum is −2.5, which is −5/2. Their product is −1.5, which is −3/2. Both agree, so the roots are right.
The reason this is worth the ink is that it survives the hard cases untouched. With 2 ± √3, the surds cancel in the sum and multiply out in the product — you get 4 and 1 without ever touching a root. It works on complex pairs for the same reason. Two multiplications, and a sign error in b cannot survive them. This page runs both on every answer before it shows you anything.
When a Is Zero and It Is Not a Quadratic
If a = 0 there is no x² term, and 2a in the denominator becomes zero. The formula does not merely give a poor answer; it cannot be applied. What you have is a linear equation — 2x − 4 = 0 and so on — and it is solved in one line without any of this.
Worth checking before you start, particularly on a question where the x² terms cancel during rearranging.
Using the Quadratic Formula Calculator
Rearrange to ax² + bx + c = 0 first, with everything on the left. Then type the three coefficients with their signs; leave b or c alone if that term is missing and they count as zero. Fractions like 3/4 work, decimals work, and the answer stays exact either way — ½x² − ⅓x is tidied to whole numbers internally so the surd comes out clean.
One habit worth forming before you reach for a quadratic formula calculator at all: write the discriminant down first. It costs one line, it tells you what shape the answer will take, and it is the only part of this process that catches a mistake before you have made it.
Open the working and you get the discriminant on its own line, the substitution before and after the signs are resolved, the square coming out of the root, and both Vieta checks with your numbers in them.
Quadratic Formula FAQ
What is the quadratic formula?
What does the discriminant tell you?
Why do I keep getting the sign of b wrong?
Is 4 plus or minus root 12 over 2 a finished answer?
How do you check the roots of a quadratic?
What if a equals zero?
Discriminant first, signs written out, the square pulled from the root — then add the roots and multiply them, and you are done.
