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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 Nature of the Solutions: Real vs. Complex 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 eitehr 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).\r\n \r\n - \n
D = b2 -4 ac < 0
\r\n The two solutions of the equation are complex and distinct.\r\n \r\n
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\n Derivation of Quadratic Formula\n
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\n Below is a step-by-step procedure to derive these solutions.\r\n
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Divide both sides of the equations by a
\r\n $$ x^2 + \\frac{b}{a} x + \\frac{c}{a} =0$$\r\n \r\n - \n
Subract c/a and add (b/2a) 2 to both sides of the equation
\r\n $$ x^2 + 2 \\left(\\frac{b}{2a}\\right) x + \\left( \\frac{b}{2a}\\right)^2 = \\left( \\frac{b}{2a}\\right)^2 - \\frac{c}{a}$$\r\n \r\n - \n
Complete the square on the left hand side of the equation. Then, simplify the right hand side
\r\n $$ \\left( x + \\frac{b}{2a} \\right)^2 = \\frac{b^2}{4a^2} - \\frac{c}{a} $$\r\n $$ \\Downarrow $$\r\n $$ \\left( x + \\frac{b}{2a} \\right)^2 = \\frac{b^2 - 4 ac}{4a^2}$$\r\n \r\n - \n
Take square root of the both sides. Then, subtract b/2a from both sides of the equation
\r\n $$ x + \\frac{b}{2a} = \\pm \\sqrt{\\frac{b^2 - 4 ac}{4a^2}} = \\pm \\frac{\\sqrt{b^2 - 4 ac}}{2a}$$\r\n $$ \\Downarrow $$\r\n $$ x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} $$\r\n \r\n
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\n MagicGraph | Solving a Quadratic Equation (Graphical Method)\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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