Quadratic Equations

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A quadratic equation in variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. This is called the standard form of a quadratic equation. The term 'quadratic'

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A real number α is called a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0. We also say that x = α is a solution of the quadratic equation. The roots are the same as the zeroes of

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A quadratic equation can have at most two roots. This is because a quadratic polynomial has degree 2, and a polynomial of degree n can have at most n zeroes (or roots when equated to zero).

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x² - 5x + 6 = 0 Factoring: x² - 3x - 2x + 6 = 0 x(x - 3) - 2(x - 3) = 0 (x - 2)(x - 3) = 0 Therefore: x - 2 = 0 or x - 3 = 0 So x = 2 or x = 3 Roots are 2 and 3.

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The quadratic formula gives the roots of ax² + bx + c = 0 as: x = (-b ± √(b² - 4ac))/2a This formula can be used when b² - 4ac ≥ 0. It is derived by completing the square method and works for all qu

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The discriminant of the quadratic equation ax² + bx + c = 0 is D = b² - 4ac. It determines the nature of roots: • If D > 0: two distinct real roots • If D = 0: two equal real roots • If D < 0: no real

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A quadratic equation ax² + bx + c = 0 has two distinct real roots when the discriminant b² - 4ac > 0. In this case, the roots are given by: x = (-b + √(b² - 4ac))/2a and x = (-b - √(b² - 4ac))/2a

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A quadratic equation ax² + bx + c = 0 has two equal (coincident) roots when the discriminant b² - 4ac = 0. In this case, both roots are equal to -b/2a. The equation can be written as a perfect square: