What is the standard form of a quadratic equation?
Standard form is ax² + bx + c = 0, with every term on one side and zero on the other. The number in front of x² is a, the number in front of x is b, and the constant is c.
The signs belong to the numbers. In x² − 5x + 6 = 0, a = 1, b = −5, and c = 6. If your equation looks like 3x² = 7x − 2, move everything to the left first: 3x² − 7x + 2 = 0. The quadratic equation solver does this rearranging for you, but it helps to know why the step is there.
Quadratic formula calculator example: x² − 5x + 6 = 0
The roots are x = 2 and x = 3. Here is the same work the calculator shows, line by line.
You can check either root by putting it back in. For x = 2: 4 − 10 + 6 = 0, so it works.
What does the discriminant tell you before you solve?
The discriminant, b² − 4ac, tells you what kind of roots to expect. A positive value gives two real roots, zero gives one repeated root, and a negative value gives no real roots.
It also tells you whether the answer will be neat. When the discriminant is a perfect square like 1, 4, or 25, the square root comes out whole and the roots are rational. Anything else leaves a radical in the answer.
| Equation | b² − 4ac | Roots |
|---|---|---|
| x² − 5x + 6 = 0 | 1 | x = 2 or x = 3 (rational) |
| x² − 2x − 1 = 0 | 8 | x = 1 ± √2 (radical) |
| x² + 2x + 5 = 0 | −16 | x = −1 ± 2i (complex) |
What if the roots have square roots or i in them?
Then the exact answer is written with √ or i, and the calculator keeps it exact instead of rounding. For x² − 2x − 1 = 0 the discriminant is 8, and √8 simplifies to 2√2, so x = (2 ± 2√2)/2 = 1 ± √2.
For x² + 2x + 5 = 0 the discriminant is −16. With the number domain set to real numbers, the result is "No real solution". Switch the setting to complex numbers and you get x = −1 ± 2i, because √−16 = 4i and (−2 ± 4i)/2 simplifies to −1 ± 2i.
When should you factor instead of using the formula?
Factor when the discriminant is a perfect square and the numbers are small, because it is usually faster. x² − 5x + 6 = 0 factors as (x − 2)(x − 3) = 0 in a few seconds.
Use the formula when factoring isn't obvious, when a is large, or when the roots turn out to be irrational. The formula always works, so it makes a good fallback. If you want to see the factored form, the factor calculator shows the ac method step by step.
A quick test helps you decide. Work out b² − 4ac first. If it is 0, 1, 4, 9, 16, 25, or another perfect square, the quadratic factors over whole numbers and factoring is worth a try. If not, go straight to the formula.
What are the most common quadratic formula mistakes?
The big one is squaring a negative b without brackets. With b = −5, b² is 25, but typing −5² into many calculators gives −25. Write (−5)² every time.
Another frequent slip is dividing only part of the top by 2a. The whole numerator, −b ± √(b² − 4ac), sits over 2a. Students also forget to move every term to one side before reading c, which changes the sign of c and every answer after it.
How do you type a quadratic into the solver?
Type it the way it is written, with ^ for the power: x^2 - 5x + 6 = 0. The quadratic calculator also accepts equations that are not yet in standard form, such as 2x^2 = 3x + 5, and any single letter as the variable.
Pasted homework text like "Solve for t:" is fine too. For equations of degree 3 to 6, the same solver uses the rational root theorem and synthetic division before applying the quadratic formula to what is left.
What do the roots mean on a graph?
The real roots are the x-intercepts of the parabola y = ax² + bx + c. They are the x-values where the curve crosses or touches the x-axis.
For y = x² − 5x + 6 the curve crosses at x = 2 and x = 3. The vertex sits halfway between them at x = 2.5, which is the same as −b/(2a) = 5/2. When the roots are complex, the parabola never reaches the x-axis at all.
Seeing this picture helps you sanity-check an answer. If a sketch shows the curve dipping below the axis twice, you should get two real roots from the formula. If you get one, recheck the discriminant.