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quadratic equation[kwä′drad·ik i′kwā·zhən]
an equation of the form ax2 + bx + c = 0, where a, b, and c are any number and are called the coefficients of the equation. A quadratic equation has two roots, which are found by the formulas
The expression D = b2 − 4ac is called the discriminant of the quadratic equation. If D > 0, then the roots of the quadratic equation are real and unequal; if D < 0, then the roots areconjugate complex numbers; if D = 0, then the roots are real andequal. The Vièta formulas x1 + x2 = − b/a and x\x2 = c/a link the roots and coefficients of a quadratic equation. The left-hand side of a quadratic equation can be expressed in the form α (x − x2) (x − x2). The function y = ax2 + bx + c is called a quadratic trinomial, and its graph is a parabola with the vertex at the point M (−b/2a; c − b2/4a) and axis of symmetry parallel to the. y-axis; the direction of the branches of the parabola coincides with the sign of a. The solution of the quadratic equation was already known in geometric form to ancient mathematicians.