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

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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

The solution of your equation

The equation you entered
Show source2x2+5x8=02\cdot{ x}^{2}+5\cdot 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=52(42(8))=25(42(8))=25(64)==25+64=89\begin{aligned}\Delta& = {5}^{2} - \left(4\cdot2\cdot\left(-8\right)\right) = 25 - \left(4\cdot2\cdot\left(-8\right)\right) = 25 - \left(-64\right) = \\ & = 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\cdot2} = \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\cdot2} = \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
    where:
    • 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 a c \Delta={ b}^{2}-4~ a~ 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 a x_1=\frac{- b-\sqrt{ \Delta}}{2~ a}
      x2=b+Δ2 a x_2=\frac{- b+\sqrt{ \Delta}}{2~ a}
    • discriminant is zero (Δ = 0) - equation has exactly one solution (so-called double root):
      h=b2 a h=\frac{- b}{2~ a}
    • discriminant is negative (Δ < 0) - equation has no solutions (so-called contradictory equation).

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