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#

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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

Calculations data - enter values, that you know here#

Function value (yy)
(the function value at single point x, often marked as f(x))
=>
Dicriminant of the quadratic function (Δ\Delta)
=>
The first root of the function (x1x_1)
=>
The second root of the function (x2x_2)
=>
X coordinate of the parabola vertex (hh)
(for this argument the function reaches its local extremum)
=>
Y coordinate of the parabola vertex (kk)
=>
Function argument (xx)
<=
Coefficient before power of two (aa)
<=
Coefficient before power of one (bb)
<=
Free parameter (cc)
<=

Result: function value (yy)#

Summary
Used formulaShow sourcey=ax2+bx+cy=a \cdot x^{2}+b \cdot x+c
ResultShow source33
Numerical resultShow source33
Result step by step
1Show source1 12+11+11~1^{2}+1 \cdot 1+1Multiply by oneAny number multiplied by one (1) gives the same number: a1=1a=aa \cdot 1 = 1 \cdot a = a
2Show source12+11+11^{2}+1 \cdot 1+1Multiply by oneAny number multiplied by one (1) gives the same number: a1=1a=aa \cdot 1 = 1 \cdot a = a
3Show source12+1+11^{2}+1+1Simplify arithmetic-
4Show source2+122+1^{2}Power of one numberThe number one (1) raised to any power gives one (1). 1n=1111n razy=11^n = \underbrace{1 \cdot 1 \cdot 1 \cdot \ldots \cdot 1}_{\text{n razy}} = 1
5Show source2+12+1Simplify arithmetic-
6Show source33ResultYour expression reduced to the simplest form known to us.
Numerical result step by step
1Show source33The original expression-
2Show source33ResultYour 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=ax2+bx+cy=a \cdot x^{2}+b \cdot x+c
    where:
    • yy - function value (the function value at single point x, often marked as f(x)),
    • xx - function argument (called also independent value),
    • aa, bb, cc - 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):
    Δ=b24 ac\Delta=b^{2}-4~a \cdot c
    where:
    • Δ\Delta - dicriminant of the quadratic function,
    • aa, bb, cc - 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=b2 ah=\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:
      x1=bΔ2 ax_1=\frac{-b-\sqrt{\Delta}}{2~a}
      x2=b+Δ2 ax_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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