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\n Quadratic Equation\n
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\n A quadratic equation is a polynomial equation that is second order in its primary variable. For example, a quadratic equation in \\(x\\) is given as:\r\n $$ a x^2 + b x +c =0$$\r\n where the coefficients \\(a \\), \\(b\\) and \\(c\\) are real numbers with the condition that \\(a \\ne 0\\).\r\n
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\n Solutions of a Quadratic Equation – Quadratic Formula\n
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\n Every quadratic equation has two solutions – which are given as:\r\n $$ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} $$\r\n The plus-minus sign (\\(\\pm\\)) indicates that the equation has two solutions. Expressed separately, the two solutions can be written as:\r\n $$x_1 = \\frac{-b + \\sqrt{b^2 - 4ac}}{2a}$$\r\n and\r\n $$x_2 = \\frac{-b - \\sqrt{b^2 - 4ac}}{2a}$$\r\n These solutions are also called the roots of the quadratic equation.\r\n
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\n Discriminant & Number of Real Solutions\n
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\n The term term inside the square root sign in the quadratic formula i.e. \\(b^2-4ac\\) is called the discriminant, and is often denoted by D. A quadratic equation can either have one solution, two distinct, real solutions, or two distinct, complex solutions. The discriminant determines the number and nature of the solution.\r\n
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D = b2 -4 ac > 0
\r\n The two solutions of the equation are real and distinct.\r\n \r\n - \n
D = b2 -4 ac = 0
\r\n The two solutions of the equation are real and indistinct (equal to each other). In other words, the equation has only one real solution.\r\n \r\n - \n
D = b2 -4 ac < 0
\r\n The two solutions of the equation are complex and distinct. In other words, the equation has no real solution.\r\n \r\n
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\n MagicGraph | Discriminant & Number of Solutions\n
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\n Graphically, a quadratic function, such as \\(y =a x^2+bx+c\\), describes a parabola when graphed in x and y. Then, the two solutions of the quadratic equation \\(a x^2+ b x +c =0\\) represent the points where this parabola intersects with the x-axis (\\(y=0\\)).\r\n
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