Quadratic Equation Calculator (Bhaskara's Formula)
Enter the coefficients a, b and c. The tool calculates the delta, finds the roots, gives the vertex of the parabola and shows every step of the solution.
Educational use: results are estimates made for study and reference. Checking the answer and deciding what to do with it is the sole responsibility of whoever uses the tool. Read the legal notice.
On this page
- How to use the calculator
- Where the formula comes from
- The quadratic formula, explained
- Worked examples
- What the delta shows on the graph
- Quick reference summary
- Mistakes that trip up a lot of people
- Questions that come up on test day
- Where quadratic equations show up
- Terms you will run into
- Limits and caution with the result
- Quadratic curiosities
- Keep going from here
How to use the calculator
A quadratic equation always has the form ax² + bx + c = 0. Before using the tool, put your equation in that form, with everything on one side and zero on the other.
- Identify the coefficients. In
2x² + 3x − 2 = 0: a = 2, b = 3, c = −2. - Type each one into its field. Use the minus sign for negative numbers. If a term is missing, its coefficient is 0 (in
x² − 9 = 0, b = 0). - Click Calculate. The calculator gives you the discriminant (delta), the roots and the vertex of the parabola.
- Open the step-by-step solution to check your work or compare it with what you did in your notebook.
If the equation is not in standard form, for example x² = 5x − 6, move everything to one side first: x² − 5x + 6 = 0.
Where the formula comes from
Solving quadratic equations is a very old problem. Babylonian clay tablets, thousands of years old, already contain problems that we would write this way today, solved by geometric methods of "completing the square".
The formula comes precisely from completing the square. Starting from ax² + bx + c = 0, you divide everything by a, move the constant term to the other side, add (b/2a)² to both sides to form a perfect square on the left and then take the square root. What is left is exactly x = (−b ± √(b² − 4ac)) ÷ 2a.
In Brazil, it is called Bhaskara's formula, after the 12th-century Indian mathematician. Historians point out that the name is a Brazilian peculiarity and that equivalent methods were known before him. In other countries, it is usually called simply the quadratic formula.
The quadratic formula, explained
The solution takes two steps: calculate the discriminant, then use the formula. In Brazil, the quadratic formula is known as Bhaskara's formula.
The delta tells you what will happen before you even find the roots:
- Δ > 0: two different real roots.
- Δ = 0: one repeated real root (both roots are equal).
- Δ < 0: no real roots. The parabola does not touch the x-axis. In the complex numbers, the roots exist and are conjugates.
Shortcuts to check the result
x₁ · x₂ = c ÷ asum and product of the roots (Vieta's formulas)
After finding the roots, add them and multiply them. If the results do not match −b/a and c/a, something went wrong. The calculator runs this check in the step-by-step solution.
The vertex of the parabola
It is the lowest point (when a > 0) or the highest point (when a < 0) of the graph. In maximum and minimum problems, it is the answer.
Worked examples
1. Two real roots: x² − 5x + 6 = 0
Checking: sum = 5 (= −b/a) and product = 6 (= c/a).
2. With a coefficient greater than 1: 2x² + 3x − 2 = 0
Notice the negative c: when a and c have opposite signs, the delta is always positive, and the equation always has two real roots.
3. Repeated root: x² + 6x + 9 = 0
A delta of zero indicates a perfect square: x² + 6x + 9 = (x + 3)². The parabola just touches the x-axis.
4. No real roots: x² + 2x + 5 = 0
This is not a calculation error. It is a legitimate answer: there is no real number that solves this equation.
5. A word problem
The area of a rectangle is 54 m². The length is 3 m more than the width. What are the dimensions?
A negative measurement does not exist, so the width is 6 m and the length is 9 m (6 × 9 = 54). In real problems, always check whether the roots make sense.
What the delta shows on the graph
Every quadratic equation is a picture of a parabola. The roots are the points where it crosses the horizontal axis, and the vertex is the turning point.
Quick reference summary
| Situation | What happens | How to recognize it |
|---|---|---|
| Δ > 0 | two different real roots | the parabola crosses the x-axis twice |
| Δ = 0 | one repeated real root | the parabola only touches the x-axis |
| Δ < 0 | no real roots | the parabola stays entirely above (a > 0) or entirely below (a < 0) the axis |
| b = 0 | incomplete equation: ax² + c = 0 | isolate x²: x = ±√(−c/a), if −c/a ≥ 0 |
| c = 0 | incomplete equation: ax² + bx = 0 | factor: x(ax + b) = 0, roots 0 and −b/a |
| a and c with opposite signs | Δ always positive | two real roots, one positive and one negative |
Incomplete equations (b = 0 or c = 0) can be solved without the quadratic formula, and it is faster that way. The calculator handles these cases normally: just type 0 for the missing coefficient.
Mistakes that trip up a lot of people
Forgetting the parentheses when squaring b
If b = −5, then b² = (−5)² = 25, not −25. Pocket calculators and spreadsheets also get this wrong when the sign is left outside the parentheses.
Getting the sign of −4ac wrong
When c is negative, −4ac becomes an addition. In 2x² + 3x − 2: −4·2·(−2) = +16. That is why it is worth writing out all the signs before you start.
Forgetting to divide everything by 2a
The denominator 2a applies to both terms of the numerator. The calculation is (−b ± √Δ) divided by 2a, not just the square root divided.
Not putting the equation in standard form first
For x² + 5 = 6x, you need to move everything to one side: x² − 6x + 5 = 0. Only then: a = 1, b = −6 and c = 5.
Accepting roots that make no sense in the problem
In problems with measurements, time or quantities, a negative root is usually discarded. Solving the equation is only half the work: the other half is interpreting the answer.
Questions that come up on test day
What is a quadratic equation?
It is an equation in which the highest power of the unknown is 2, in the form ax² + bx + c = 0, with a different from zero.
How do I use the quadratic formula?
Calculate the delta (b² − 4ac) and plug it into x = (−b ± √Δ) ÷ 2a. The ± sign gives the two roots.
What does a negative delta mean?
It means there are no real roots: the graph does not cross the x-axis. In the complex numbers, there are two roots, in the form a ± bi.
What happens if a is zero?
The equation stops being quadratic and becomes a linear equation (bx + c = 0), with the solution x = −c/b. The calculator warns you when this happens.
How do I solve an incomplete equation without the quadratic formula?
If b = 0, isolate x² and take the square root (x = ±√(−c/a)). If c = 0, factor: x(ax + b) = 0, and the roots are 0 and −b/a.
What is the vertex of the parabola?
It is the minimum point (if a > 0) or maximum point (if a < 0) of the function. Its coordinates are xᵥ = −b/2a and yᵥ = −Δ/4a.
How can I check that the roots are right?
Add them and multiply them. The sum should be −b/a and the product c/a. You can also substitute each root into the original equation and see whether it gives zero.
Why does the result have two roots?
Because when a parabola crosses the x-axis, it usually does so at two points. The ± sign in the formula expresses that.
Can I enter decimal or negative coefficients?
Yes. Use the minus sign for negative numbers and a period or comma for decimals.
Does the calculator solve equations like x² = 5x − 6?
Yes, as long as you rearrange them first: x² − 5x + 6 = 0. Then just enter a = 1, b = −5 and c = 6.
Where quadratic equations show up
- Physics: motion with constant acceleration leads to quadratic equations, such as the height of a ball thrown upward over time.
- Geometry: finding the sides of a shape from its area, as in the rectangle example.
- Optimization: the vertex of the parabola gives the maximum or minimum value of a quantity, such as the largest area you can fence in with a fixed length of wire mesh.
- Engineering and design: suspension cables, satellite dishes and headlights use the shape of the parabola.
- Statistics and economics: simple models of cost and revenue curves use quadratic functions.
Terms you will run into
- Coefficient
- A number that multiplies the unknown. In the equation 3x² + 2x − 1 = 0, the coefficients are a = 3, b = 2 and c = −1.
- Discriminant (Δ)
- The value of b² − 4ac, which tells you how many real roots the equation has.
- Root
- A value of x that makes the equation true. Also called a solution or a zero of the function.
- Parabola
- The curve that represents a quadratic function on a graph.
- Vertex
- The maximum or minimum point of the parabola.
- Concavity
- The direction in which the parabola opens: upward (a > 0) or downward (a < 0).
- Complex number
- A number with a real part and an imaginary part, such as −1 + 2i. It solves equations that have no real roots.
Limits and caution with the result
The calculator works with double-precision decimal arithmetic (about 15 digits). For very large or very small coefficients, or a delta very close to zero, small rounding errors can appear. When the delta is practically zero, the tool treats it as zero and reports a repeated root.
Complex roots are shown in the form a ± bi for educational purposes. If your course only deals with real numbers, the correct answer in that case is "no real roots". If the result is used in a problem with measurements, always check that it makes sense in context.
Bottom line: this tool is educational. Results do not replace your own checking, and how you use them is entirely your responsibility. More details in the legal notice.
Quadratic curiosities
- "Quadratic" comes from "square". The name comes from the x² term, which is the area of a square with side x.
- Parabolas in everyday life: the stream of water from a drinking fountain, a basketball on its way to the hoop and the path of a thrown stone are all approximately parabolas (ignoring air resistance).
- The golden ratio is the positive root of the equation x² − x − 1 = 0: Δ = 5, x = (1 + √5) ÷ 2 = 1.618...
- Complex roots seem abstract, but they are essential in electronics and in wave physics.