Quadratic equations solver calculator
Calculator finds out solution of quadratic equation given in general ax²+bx+c=0 form.

Beta version#

This online calculator is currently under heavy development. It may or it may NOT work correctly.
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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

Input equation, which you want to solve#

Parameters of the ax2 + bx + c = 0 equation
Coefficient a
(just before x2)
Coefficient b
(just before x)
Free parameter c
Unknown variable
(the variable, which we're searching for)

The solution of your equation#

The equation you entered
Show source2 x2+5 x8=02~x^{2}+5~x-8 = 0
The solution of the equation
Show sourcex{54894,54+894} x \in \left\{\frac{-5}{4}-\frac{\sqrt{89}}{4}, \frac{-5}{4}+\frac{\sqrt{89}}{4}\right\}

The solution step-by-step#

I. We calculatate discriminant of the quadratic equation Δ\Delta:
Δ=52428=25428=25+842=25+322=25+64=89\begin{aligned}\Delta& = 5^{2}-4 \cdot 2 \cdot -8 = 25-4 \cdot 2 \cdot -8 = 25+8 \cdot 4 \cdot 2 = 25+32 \cdot 2 = 25+64 = 89\end{aligned}
II. Delta is positive (Δ > 0), so equation has two solutions (roots).
The first solution is::
x1=58922=5894=54894\begin{aligned}x_1& = \frac{-5-\sqrt{89}}{2 \cdot 2} = \frac{-5-\sqrt{89}}{4} = \frac{-5}{4}-\frac{\sqrt{89}}{4}\end{aligned}The second solution is::
x2=5+8922=5+894=54+894\begin{aligned}x_2& = \frac{-5+\sqrt{89}}{2 \cdot 2} = \frac{-5+\sqrt{89}}{4} = \frac{-5}{4}+\frac{\sqrt{89}}{4}\end{aligned}

Some facts#

  • The quadratic equation is an equation that can be presented in the form:
    a x2+b x+c=0a~x^2 + b~x + c = 0
    • a, b, c - constant parameters, these are numbers that we know,
    • x - unknown variable, it's a number, which we search for.
  • Quadratic equation can have one solution, two solutions or do not have solutions.
  • The universal method of solving quadratic equations uses discriminant of the quadratic polynomial (so-called delta):
    Δ=b24 ac\Delta=b^{2}-4~a \cdot c
  • When we calculate the discriminant, three scenarios are possible:
    • discriminant is positive (Δ > 0) - equation has two different solutions (two different roots):
      x1=bΔ2 ax_1=\frac{-b-\sqrt{\Delta}}{2~a}
      x2=b+Δ2 ax_2=\frac{-b+\sqrt{\Delta}}{2~a}
    • discriminant is zero (Δ = 0) - equation has exactly one solution (so-called double root):
      h=b2 ah=\frac{-b}{2~a}
    • discriminant is negative (Δ < 0) - equation has no solutions (so-called contradictory equation).

See also#

If you are interested in solving mathematical equations, check out our other calculators:
  • Linear equation solver - see how to solve a linear equation in the form ax+b=0ax + b = 0 step by step,
  • Quadratic equation solver - see how to solve quadratic equation in the form ax2+bx+c=0ax ^ 2 + bx + c = 0 using the so-called delta scheme,
  • General equation solver - if you don't know which solving method should be applied to your equation, just give us the left and right side and we will try to solve it for you.

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