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

# Beta version

BETA TEST VERSION OF THIS ITEM
This online calculator is currently under heavy development. It may or it may NOT work correctly.
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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 source$2\cdot{ x}^{2}+5\cdot x - 8 = 0$ The solution of the equation Show source$x \in \left\{\frac{-5}{4}-\frac{\sqrt{89}}{4}, \frac{-5}{4}+\frac{\sqrt{89}}{4}\right\}$

# The solution step-by-step

I. We calculatate discriminant of the quadratic equation $\Delta$.
\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:
\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:
\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:
$a~x^2 + b~x + c = 0$
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):
$\Delta={ b}^{2}-4~ a~ c$
• When we calculate the discriminant, three scenarios are possible:
• discriminant is positive (Δ > 0) - equation has two different solutions (two different roots):
$x_1=\frac{- b-\sqrt{ \Delta}}{2~ a}$
$x_2=\frac{- b+\sqrt{ \Delta}}{2~ a}$
• discriminant is zero (Δ = 0) - equation has exactly one solution (so-called double root):
$h=\frac{- b}{2~ a}$
• discriminant is negative (Δ < 0) - equation has no solutions (so-called contradictory equation).

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