Fungrim entry: fc2582

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \left[a {z}^{2} + b z + c\right] = \left\{\frac{-b + \sqrt{{b}^{2} - 4 a c}}{2 a}, \frac{-b - \sqrt{{b}^{2} - 4 a c}}{2 a}\right\}

Assumptions:

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \ne 0

TeX:

\mathop{\operatorname{zeros}\,}\limits_{z \in \mathbb{C}} \left[a {z}^{2} + b z + c\right] = \left\{\frac{-b + \sqrt{{b}^{2} - 4 a c}}{2 a}, \frac{-b - \sqrt{{b}^{2} - 4 a c}}{2 a}\right\}

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; c \in \mathbb{C} \;\mathbin{\operatorname{and}}\; a \ne 0

Definitions:

Fungrim symbol	Notation	Short description
`Zeros`	$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$	Zeros (roots) of function
`Pow`	${a}^{b}$	Power
`CC`	$\mathbb{C}$	Complex numbers
`Sqrt`	$\sqrt{z}$	Principal square root

Source code for this entry:

Entry(ID("fc2582"),
    Formula(Equal(Zeros(Add(Add(Mul(a, Pow(z, 2)), Mul(b, z)), c), ForElement(z, CC)), Set(Div(Add(Neg(b), Sqrt(Sub(Pow(b, 2), Mul(Mul(4, a), c)))), Mul(2, a)), Div(Sub(Neg(b), Sqrt(Sub(Pow(b, 2), Mul(Mul(4, a), c)))), Mul(2, a))))),
    Variables(a, b, c),
    Assumptions(And(Element(a, CC), Element(b, CC), Element(c, CC), NotEqual(a, 0))))

Topics using this entry

Square roots