Quadratic equation

In mathematics, a quadratic equation is a polynomial equation of the second degree. The generalized form is

Quadratic formula

The quadratic formula explicitly gives the solutions of a quadratic equation in terms of the coefficients a, b and c, which we temporarily assume to be real (but see below for generalizations) with a being non-zero. These solutions are also called the roots of the equation. The formula reads

Related Topics:
Real - Roots

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x_{1,2}=rac{-b pm sqrt {b^2-4ac }}{2a}.

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An alternative form sometimes encountered is given by

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x_{1,2}=rac{2c}{-b pm sqrt {b^2-4ac }}.

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x_{1,2}=rac{-(10) pm sqrt {(10)^2-4(8)(-33) }}{2(8)}

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This gives the solutions x_1=3/2 and x_2=-11/4.}}

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The term b2 − 4ac is called the discriminant of the quadratic equation, because it discriminates between three qualitatively different cases:

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• If the discriminant is zero then there is a repeated solution x, and this solution is real. Geometrically, this means that the parabola described by the quadratic equation touches the x-axis in a single point.
• If the discriminant is positive, then there are two different solutions x, both of which are real. Geometrically, this means that the parabola intersects the x-axis in two points. Furthermore, if the discriminant is a perfect square, the roots are rational numbers -- in other cases they may be quadratic irrationals.
• If the discriminant is negative, then there are two different solutions x, both of which are complex numbers. The two solutions are complex conjugates of each other. In this case, the parabola does not intersect the x-axis at all.

Note that when computing roots numerically, the usual form of the quadratic formula is not ideal. See Loss of significance for details.

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