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

# Beta version#

BETA TEST VERSION OF THIS ITEM
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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).

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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