Quadratic Equation Calculator

Typical Use Cases

You often encounter quadratic equations in everyday life, even if you don't realize it. For example:

• Home Renovation: When laying floor tiles, if the room's length and width follow a quadratic relationship, you need to solve an equation to determine the cutting dimensions.
• Projectile Motion: The trajectory of a basketball shot is a parabola. To calculate when the ball will hit the ground, you use a quadratic equation.
• Personal Finance: Monthly mortgage payment formulas also contain quadratic terms that require solving equations when simplified.

You don't need to calculate these by hand. Just use our calculator, enter the coefficients, and get the answer instantly.

How to Use

1. Open our calculator page, and you will see three input fields labeled a, b, and c, corresponding to the coefficients in the equation ax² + bx + c = 0.
2. Enter the value of a in the first box (e.g., 2), b in the second box (e.g., -4), and c in the third box (e.g., -6). Note that negative numbers must include a minus sign.
3. Click the "Calculate" button below.
4. The result area will display: the value of the discriminant Δ, the number of roots, and the two roots x₁ and x₂. If Δ < 0, it will display them in complex number format.
5. You can also directly copy the results or clear the fields to enter new values.

Complete Example

Let's calculate one together: 2x² - 4x - 6 = 0.

• Enter 2 in the a input box, -4 for b, and -6 for c, then click calculate.
• The calculator computes internally: Discriminant Δ = b² - 4ac = (-4)² - 4×2×(-6) = 16 + 48 = 64
• √Δ = 8, then the two roots are: x₁ = [4 + 8] / (2×2) = 12 / 4 = 3; x₂ = [4 - 8] / 4 = -4 / 4 = -1
• The result page shows: Δ = 64 > 0, two distinct real roots: 3 and -1. You can immediately verify this with the original equation: when x=3, 2×9 - 12 - 6 = 0, which is correct.

More Examples

Example 2 (Repeated Root): x² + 2x + 1 = 0, meaning a=1, b=2, c=1. The calculation yields Δ = 0, and the roots x₁ = x₂ = -1. The result page will indicate "two equal real roots".

Example 3 (No Real Roots): x² + x + 1 = 0, meaning a=1, b=1, c=1. Δ = 1 - 4 = -3 < 0. The calculator will output two complex roots: (-1 ± i√3)/2. This is very useful for college engineering calculations.

How to Interpret the Results

• Δ > 0: The equation has two distinct real roots. The parabola opens upward or downward and intersects the x-axis at two points.
• Δ = 0: The equation has two equal real roots (a repeated root). The vertex of the parabola lies exactly on the x-axis.
• Δ < 0: The equation has no real roots, but has a pair of complex conjugate roots. The parabola lies completely above or below the x-axis and does not touch it.

Note: Our calculator will display the results in reduced fraction form (if the coefficients are integers and the roots are rational) or as decimals. Complex roots will be written in the a ± bi format.

Common Mistakes

• Entering coefficients in the wrong order: Ensure the input order is a, b, c. Do not enter them as b, a, c or c, b, a.
• Missing negative signs: If the equation is 2x² - 4x + 6 = 0, b should be entered as -4, not 4; c should be 6.
• a cannot be 0: If a=0, it becomes a linear equation, and this tool does not apply. If a=0 is entered, the result will prompt "This is not a quadratic equation".
• Expecting exact fractions for irrational inputs: When coefficients are irrational numbers (like √2), it is recommended to convert them to decimals before inputting; the result will show a decimal approximation.
• Forgetting to check the discriminant's sign: Before looking at the roots, check whether Δ is positive or negative to quickly determine if there are real roots.

Important Notes

• This calculator only handles quadratic equations (the highest degree term is x²). It cannot solve cubic or higher-degree equations.
• Coefficients a, b, and c support integers, decimals, and negative numbers, but a must be a non-zero real number. If a=0, the tool will return an error.
• Floating-point calculations have a very small rounding error (usually less than 1e-10), but this does not affect most applications.
• The output format for complex roots is in rectangular coordinates (a+bi); polar coordinate forms are not provided.
• This tool is based on the quadratic formula, which is completely consistent with textbook methods.

Frequently Asked Questions (FAQ)

1. Why are there no real roots when Δ < 0? The square root of a negative number is undefined in the real number system, so there are no real solutions. However, solutions exist in the complex number system, and the calculator will provide them in complex form.
2. How do I input fractional coefficients (e.g., a=1/2)? Simply use the decimal 0.5. If you want to keep fractional results, you can calculate them manually before inputting, but the results will still be displayed as decimals.
3. Is the order of the two roots in the result fixed? Yes, x₁ is the larger root (when Δ≥0), and x₂ is the smaller one. For complex roots, x₁ has a real part of (-b)/(2a) and a positive imaginary part.
4. Can this calculator solve cubic equations? No, it can only solve quadratic equations. Please use a dedicated tool for cubic equations.
5. Why is there a slight difference between my roots and the manual calculation? It might be due to floating-point rounding. For example, using 0.333333 instead of 1/3 will result in a minor difference. It is recommended to input exact decimals.
6. Can coefficients be letters? No. This tool only accepts numerical inputs and does not support symbolic algebra.

Now you can try your own numbers in the calculator above.