Enter the three coefficients (a, b, c) of the quadratic equation.
Be sure to provide the correct sign (+ or –) for each coefficient.
The quadratic equation must be in standard form, as shown below:
Press Enter or Return on the keyboard or keypad to find the solution (roots) of the quadratic equation.
Input Format
Each coefficient should be in one of the following formats.
You can mix the formats:
Integer:31457Decimal:+9862.314Scientific Notation:2.7E+2Your entered coefficients should look like one of these:8 0 33.4 14 5.56e-35 2.7E+2 3.718

First Quadratic Root

This output field shows the first quadratic root, from the (+) version of the quadratic formula:
A quadratic equation may have one real root, two real roots, or two complex roots.
When the root or roots are real, they are the x-intercepts of the quadratic equation, or in other words,
the point or points where the quadratic equation intercepts the x-axis.
When the roots are complex, the quadratic equation does not intercept the x-axis.
Real Root
A root whose value is a Real number, such as:
3-5.3862.3e-4Complex Root
A root whose value is a Complex number. Complex numbers have a Real number part and an Imaginary number part, such as:
3 + (5)i-5.386 - (8.3)i2.3e-4 + (5.3e-3)i
Complex roots always occur in pairs, and are always complex conjugates of each other.

Second Quadratic Root

This output field shows the second quadratic root, from the (–) version of the quadratic formula:
A quadratic equation may have one real root, two real roots, or two complex roots.
When the root or roots are real, they are the x-intercepts of the quadratic equation, or in other words,
the point or points where the quadratic equation intercepts the x-axis.
When the roots are complex, the quadratic equation does not intercept the x-axis.
Real Root
A root whose value is a Real number, such as:
3-5.3862.3e-4Complex Root
A root whose value is a Complex number. Complex numbers have a Real number part and an Imaginary number part, such as:
3 + (5)i-5.386 - (8.3)i2.3e-4 + (5.3e-3)i
Complex roots always occur in pairs, and are always complex conjugates of each other.

Vertex Minimum or Maximum

This output field shows the x,y point that is the Minimum or Maximum of the quadratic equation,
also known as the Vertex of the quadratic equation.
The x and y coordinates of the Vertex of the quadratic equation are defined as:
The upper case Greek delta (Δ) stands for the Discriminant of the quadratic equation. Δ is the
expression inside the square root sign of the quadratic formula, and is defined as:
Minimum or Maximum
The graph of all quadratic functions has the same general shape, called a parabola. The location and size of the parabola,
and whether it opens up or down, depend on the values of the coefficients a, b, and c:
If a > 0, the parabola has a minimum point and opens upward.
If a < 0, the parabola has a maximum point and opens downward.

Discriminant

This output field shows the Discriminant of the quadratic equation. The Discriminant derives from the quadratic formula:
The expression inside the square root sign is called the Discriminant of the quadratic equation,
and is represented with an upper case D or an upper case Greek delta (Δ):
If Δ > 0 there are two different real roots.
If Δ = 0 there is exactly one real root.
If Δ < 0 there are two different complex roots.

Focus

This output field shows the Focus of the parabola. The Focus is defined as:
The upper case Greek delta (Δ) stands for the Discriminant of the quadratic equation. Δ is the
expression inside the square root sign of the quadratic formula, and is defined as:

Focal Length

This output field shows the Focal Length of the parabola. The Focal Length is defined as:

Axis of Symmetry

This output field shows the Axis of Symmetry of the parabola. A parabola has a single Axis of Symmetry, which passes
through its Focus and is perpendicular to its Directrix. The point of intersection of the Axis of Symmetry and the parabola
is the Vertex. Numerically, the Axis of Symmetry is defined as:

Directrix

This output field shows the Directrix of the parabola. The Directrix is a line perpendicular to the axis of
symmetry of the parabola such that the Vertex of the parabola is at the midpoint, along the axis of symmetry,
between the Focus and the Directrix. Numerically, the Directrix is defined as:
The upper case Greek delta (Δ) stands for the Discriminant of the quadratic equation. Δ is the
expression inside the square root sign of the quadratic formula, and is defined as:

About the Quadratic Formula Solver

TIP:Entry Errors
If the data you enter into any field contains errors for that field, the converter will notify you.
In that case, simply re-enter valid data in the field of your choice and press Enter or Return.
TIP:Decimal Precision
The decimal precision of the converter is 9 significant digits, from left to right. Digits after the
9 left-most digits, are not significant.
The Numeric Range of the converter is:
+1.797e+308 ... -1.797e+308+1.000e-311 ... -1.000e-311TIP:Copy and Paste
You can Copy and Paste any of the fields by selecting the text in the field,
then pressing Control-C to copy, and Control-V to paste.

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