Quadratic Equation Calculator

1. Write your equation in the form a x^2 + b x + c = 0.
2. Enter the three coefficients a, b and c.
3. Read the roots, whether real or complex.
4. Check the discriminant and the vertex of the parabola.

Here is what each result means:

A quadratic equation is a second-degree equation of the form a x^2 + b x + c = 0, where a is not zero. Its graph is a parabola, and the roots are the points where that parabola meets the x-axis. A quadratic has at most two roots, which may be two real numbers, one repeated real number, or a pair of complex numbers.

Quadratics appear across science and daily life: the path of a thrown ball, the area of a rectangle with a fixed perimeter, and many optimization problems all lead to one. Solving it means finding the x values that satisfy the equation, and the quadratic formula does that for every case.

The calculator uses the quadratic formula, which solves any quadratic directly from its coefficients.

1. Compute the discriminant D = b^2 - 4ac.
2. If D is positive there are two real roots; if zero, one repeated root; if negative, two complex roots.
3. Substitute into the formula and simplify for each root.

For higher powers, the cube root calculator and other tools handle roots that a quadratic does not.

Solve x^2 - 5x + 6 = 0, so a is 1, b is -5 and c is 6.

Calculation: the discriminant is (-5)^2 - 4(1)(6) = 25 - 24 = 1, which is positive. The roots are x = (5 +/- sqrt(1)) / 2, giving x = 3 and x = 2. The vertex sits at (2.5, -0.25), the lowest point of the parabola.

The sign of the discriminant b^2 - 4ac decides the number and type of roots before you solve.

This single number is often all you need when a problem asks only whether real solutions exist.

The formula always works, but other methods can be quicker in special cases.

This tool uses the formula, so it handles messy or complex roots that do not factor neatly.

The Leading Coefficient A

It sets how wide the parabola is and which way it opens. A larger size makes a narrower curve.

The Discriminant

The combination b squared minus 4ac decides whether the roots are real, repeated or complex.

The Constant C

It is where the parabola crosses the y-axis and shifts the whole curve up or down.

Checking Homework

Confirm the roots you found by hand, including tricky complex answers.

Physics and Motion

Solve for the time a projectile lands, which often forms a quadratic.

Design and Area

Find a dimension when an area or product condition gives a quadratic.

1. Sign Errors on B and C

Move every term to one side first, then read the signs carefully into the formula.

2. Forgetting the Plus-or-minus

The square root gives two values. Missing one loses a root.

3. Dividing Only Part by 2a

The whole numerator, not just the root, is divided by 2a.

4. Ignoring Complex Roots

A negative discriminant still has solutions; they are complex, not nonexistent.

5. Treating a Linear Equation as Quadratic

If a is zero it is not a quadratic. Solve bx + c = 0 instead.

The formula is exact; only the displayed decimals are rounded.

Put it in the form a x squared plus b x plus c equals zero, then apply the quadratic formula x equals minus b plus or minus the square root of b squared minus 4ac, all over 2a. This tool does that and returns both roots.

It is b squared minus 4ac. If it is positive there are two real roots, if zero one repeated root, and if negative two complex roots. It tells you the type of solution before you solve.

Yes. When the discriminant is negative, the parabola never crosses the x-axis, so there are no real roots. The two solutions are complex numbers instead.

When the discriminant is negative, the roots take the form of a real part plus or minus an imaginary part times i, where i is the square root of minus one. They come as a conjugate pair.

Then it is not a quadratic but a linear equation, b x plus c equals zero, with a single root x equals minus c over b. The calculator handles this case for you.

It is the turning point of the parabola, at x equals minus b over 2a. The y value there is the minimum if a is positive or the maximum if a is negative.

Factoring is quick when the roots are simple whole numbers. The quadratic formula always works, including for messy or complex roots, which is why this tool uses it.

Results are rounded to four decimals for display. A root that is really a whole number may show a tiny rounding difference in the last place.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Quadratic equation (Wikipedia).
• Quadratic formula explained (Maths Is Fun).
• The discriminant (Maths Is Fun).