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

# Beta version#

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
Feel free to send any ideas and comments !

# 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 formulaShow source$y=a \cdot x^{2}+b \cdot x+c$
ResultShow source$3$
Numerical resultShow 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.

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