Quadratic equations solver calculator
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.
You CAN try to use it. You CAN even get the proper results.
However, please VERIFY all results on your own, as the level of completion of this item is NOT CONFIRMED.
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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) x

# 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).

# 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 $ax + b = 0$ step by step,
• Quadratic equation solver - see how to solve quadratic equation in the form $ax ^ 2 + bx + c = 0$ 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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