Quadratic Formula Calculator With Steps and Roots

Creator: Abdullah ArshadReviewer: Parker Rowland

Type in your a, b, and c values and this quadratic formula calculator returns the roots, the discriminant, and every step in between. It handles two real roots, one repeated root, and complex roots, so any equation shaped like ax² + bx + c = 0 gets solved the same way you would on paper, just faster and without the sign slips.

Below the tool you will find worked examples using the exact problems students paste in, plus a quick method for equations that have terms on both sides.

What Does a Quadratic Formula Calculator Do?

It solves any equation of the form ax² + bx + c = 0 for x. You enter the three coefficients, and the tool plugs them into the quadratic formula, works out the discriminant, and gives you the roots with each step shown. No factoring, no guessing, no graphing needed.

The formula it uses is:

x = ( −b ± √(b² − 4ac) ) / 2a

Every quadratic can be solved with this one formula. That is what makes it reliable when factoring gets messy or the numbers do not split cleanly.

How Do You Solve a Quadratic Equation Step by Step?

Four steps, every time:

1. Write it in standard form. Move everything to one side so the equation equals zero: ax² + bx + c = 0.
2. Identify a, b, and c. These are the coefficients of x², x, and the constant.
3. Find the discriminant. Calculate b² − 4ac. This single number tells you how many solutions you get.
4. Plug into the formula. Substitute a, b, and c, then simplify to get your roots.

The calculator above does steps 3 and 4 for you and shows the working. You still need step 1 and 2, which is why the section on equations with terms on both sides matters. If you also want the line form of related equations or the axis the parabola is symmetric around, the slope intercept form calculator pairs well with this one for graphing work.

What Is the Discriminant and Why Does It Matter?

The discriminant is the part under the square root: b² − 4ac. You do not need to finish the whole formula to know what kind of answer you will get. The discriminant tells you upfront.

So if your discriminant comes out negative, you already know the answer involves i (the imaginary unit) before you finish. A square root calculator helps when the discriminant is a positive number that is not a perfect square and you want the surd form.

Worked Examples Using the Quadratic Formula

These are the exact kinds of problems people paste into the tool. Each one is solved in full so you can match your own steps against them.

Example 1: 2v² + 7v + 5 = 0
a = 2, b = 7, c = 5
Discriminant: 7² − 4(2)(5) = 49 − 40 = 9
√9 = 3, so x = ( −7 ± 3 ) / 4
x = −4/4 = −1 and x = −10/4 = −2.5

Example 2: 9a² − 18a + 5 = 0
a = 9, b = −18, c = 5
Discriminant: (−18)² − 4(9)(5) = 324 − 180 = 144
√144 = 12, so x = ( 18 ± 12 ) / 18
x = 30/18 = 5/3 and x = 6/18 = 1/3

Example 3: x² + 4x + 5 = 0 (complex roots)
a = 1, b = 4, c = 5
Discriminant: 4² − 4(1)(5) = 16 − 20 = −4
Since it is negative, √−4 = 2i
x = ( −4 ± 2i ) / 2 = −2 ± i

How Do You Solve Equations With Terms on Both Sides?

Many homework problems do not start in standard form. You will see something like −b² − 18b − 8 = −6b², with x² terms on both sides. The calculator needs standard form first, so you move everything to one side.

Example: −b² − 18b − 8 = −6b²
Add 6b² to both sides: −b² + 6b² − 18b − 8 = 0
Combine: 5b² − 18b − 8 = 0
Now a = 5, b = −18, c = −8
Discriminant: (−18)² − 4(5)(−8) = 324 + 160 = 484
√484 = 22, so x = ( 18 ± 22 ) / 10
x = 40/10 = 4 and x = −4/10 = −0.4

The rule is simple: get one side to zero, combine like terms, then read off a, b, and c. Once it is in standard form, the tool takes over.

What Are the Roots of a Quadratic Equation?

The roots are the x values that make the equation equal zero. On a graph, they are the points where the parabola crosses the x-axis. Geometry problems often lead to quadratics too. Finding the side of a shape from its area, for example, means solving a squared equation, which is why area tools like the area of a rectangle calculator pair naturally with this one.

A quadratic can cross twice (two real roots), touch once (one repeated root), or never touch (complex roots). The discriminant decides which case you are in, and the sign of a tells you whether the parabola opens up or down.

When Should You Use the Quadratic Formula Instead of Factoring?

Factoring is faster when the numbers are small and split cleanly. The quadratic formula wins when they do not. Use the formula when:

• The equation does not factor into whole numbers
• The roots are decimals, fractions, or surds
• The discriminant is negative and the answer is complex
• You want a guaranteed method that works every single time

Factoring can fail you. The formula never does. That is why it is the fallback method taught in every algebra class.

Common Mistakes to Avoid

• Sign errors on b. If b is negative, −b becomes positive. This trips people up constantly.
• Forgetting standard form. The formula only works when the equation equals zero. Move everything over first.
• Dropping the ±. There are two roots. The plus and minus give you both.
• Misreading a negative discriminant. A negative under the root means complex answers, not “no answer.”

Frequently Asked Questions

Can this calculator handle complex or imaginary roots?
Yes. When the discriminant is negative, it returns the roots in a ± bi form instead of stopping.

What if a equals zero?
Then it is not a quadratic, it is a linear equation. The x² term disappears and the quadratic formula does not apply.

Does it show the steps?
Yes. It displays the substitution, the discriminant, and how each root is found, so you can follow the working, not just copy the answer.

Can I enter decimals or fractions as coefficients?
Yes, decimals work directly. For fractions, convert to a decimal first or clear the fraction by multiplying the whole equation through.

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