Calculator helpful during common operations related to quadratic function such as calculating value at given point, calculating discriminant or finding out function roots.

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Symbolic algebra

ⓘ Hint: This calculator supports symbolic math. You can enter numbers, but also symbols like a, b, pi or even whole math expressions such as (a+b)/2. If you still don't sure how to make your life easier using symbolic algebra check out our another page: Symbolic calculations

What do you want to calculate today?#

Choose a scenario that best fits your needs

I know function argument (

x

) and quadratic function coefficients (

a

b

c

) and want to calculate function value (

y

)I know function argument (

x

), coefficient before power of two (

a

) and function zero points (

x_1

x_2

) and want to calculate function value (

y

)I know function argument (

x

), coefficient before power of two (

a

) and coordinates of the parabola vertex (

h

k

) and want to calculate function value (

y

)I know quadratic function coefficients (

a

b

c

) and want to calculate dicriminant of the quadratic function (

\Delta

)I know coefficient before power of one (

b

), coefficient before power of two (

a

) and dicriminant of the quadratic function (

\Delta

) and want to calculate the first root of the function (

x_1

)I know coefficient before power of one (

b

), coefficient before power of two (

a

) and dicriminant of the quadratic function (

\Delta

) and want to calculate the second root of the function (

x_2

)I know coefficient before power of one (

b

) and coefficient before power of two (

a

) and want to calculate x coordinate of the parabola vertex (

h

)I know dicriminant of the quadratic function (

\Delta

) and coefficient before power of two (

a

) and want to calculate y coordinate of the parabola vertex (

k

)

Calculations data - enter values, that you know here#

Function value ( $y$ ) (the function value at single point x, often marked as f(x))		=>
Dicriminant of the quadratic function ( $\Delta$ )		=>
The first root of the function ( $x_1$ )		=>
The second root of the function ( $x_2$ )		=>
X coordinate of the parabola vertex ( $h$ ) (for this argument the function reaches its local extremum)		=>
Y coordinate of the parabola vertex ( $k$ )		=>
Function argument ( $x$ )		<=
Coefficient before power of two ( $a$ )		<=
Coefficient before power of one ( $b$ )		<=
Free parameter ( $c$ )		<=

Result: function value ( $y$ )#

Summary

Used formula

Show source

y=a \cdot x^{2}+b \cdot x+c

Result

Show source

3

Numerical result

Show source

3

Result step by step

1	Show source $1~1^{2}+1 \cdot 1+1$	Multiply by one	Any number multiplied by one (1) gives the same number: $a \cdot 1 = 1 \cdot a = a$
2	Show source $1^{2}+1 \cdot 1+1$	Multiply by one	Any number multiplied by one (1) gives the same number: $a \cdot 1 = 1 \cdot a = a$
3	Show source $1^{2}+1+1$	Simplify arithmetic	-
4	Show source $2+1^{2}$	Power of one number	The number one (1) raised to any power gives one (1). $1^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{\text{n razy}} = 1$
5	Show source $2+1$	Simplify arithmetic	-
6	Show source $3$	Result	Your expression reduced to the simplest form known to us.

Numerical result step by step

1	Show source $3$	The original expression	-
2	Show source $3$	Result	Your expression reduced to the simplest form known to us.

Some facts#

The quadratic function is a function that can be prepresented in the form:

$y=a \cdot x^{2}+b \cdot x+c$
where:
- $y$ - function value (the function value at single point x, often marked as f(x)),
- $x$ - function argument (called also independent value),
- $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).
The graph of the quadratic function is parabola. Depending on the coefficient value at the second power (a), the following scenarios are possible:
- when the coefficient on the second power is positive (a> 0) - the parabola's arms are directed upwards,
- when the coefficient on the second power is negative (a < 0) - the parabola arms are directed downwards,
- in the case when the coefficient on the second power is equal to zero (a = 0) - the quadratic function reduces to linear function.
A square function can have one, two, or have no zero points. To check the number of zero places (sometimes also called roots), we can calculate the discriminant of a quadratic function (colloquially called delta):

$\Delta=b^{2}-4~a \cdot c$
where:
- $\Delta$ - dicriminant of the quadratic function,
- $a$ , $b$ , $c$ - quadratic function coefficients (numbers just before x², x and free parameter).
then the following scenarios are possible:
- discriminant is negative (Δ <0) - the function has no roots, the graph of the function is a parabola, which is located entirety above the OX axis or under the OX axis,
- discriminant is equal to zero (Δ = 0) - the function has exactly one root, the graph of the function is a parabola whose vertex lies on the OX axis:
  
  $h=\frac{-b}{2~a}$
- discriminant is positive (Δ> 0) - the function has two different roots, the function graph is a parabola, whose arms cross the OX axis:
  
  $x_1=\frac{-b-\sqrt{\Delta}}{2~a}$
  
  $x_2=\frac{-b+\sqrt{\Delta}}{2~a}$
A quadratic function is a special case of polynomial function in which the order is 2.