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

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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 source $2~x^{2}+5~x-8 = 0$
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}-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::

\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::

\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:

$a~x^2 + b~x + c = 0$
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):

$\Delta=b^{2}-4~a \cdot c$
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).