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Transcript
Pre-calculus: Quadratic Roots What is a quadratic equation? A Quadratic equation is a statement for equating a polynomial of degree 2 to zero: P(x) = 0 deg[P(x)] = 2 Examples: x 2 + 2x + 1 = 0 3x 2 + 1 − 5 = 0 What is a root of a quadratic equation? A number, α is a root of a quadratic equation if and only if P ( α ) = 0. What is the difference between the term ‘root’ and ‘zero’? The term ‘root’ is for equations while the term ‘zero’ is for a polynomial. We say α is a root of a polynomial equation P ( x ) = 0. We say α is a zero of the polynomial P ( x ) such that P ( α ) = 0. How many roots does a quadratic equation has? According to the fundamental theorem of algebra, a quadratic equation has at most 2 roots (including real and imaginary roots). What is the general form of a quadratic equation? A x2 + Bx + C = 0. What is the determinant of a quadratic equation? What does it tell? The determinant of a quadratic equation is D = B2 - 4 A C. It tells the nature of the roots: determinant \ nature number of roots nature of roots exactly zero 1 real, double greater than zero 2 both are real less than zero 2 both are imaginary Pre-calculus: Quadratic Roots Why the determinant of a quadratic equation tells the nature of the roots? Since the general form of a quadratic equation, A x2 + B x + C = 0 Can be represented as A { x + [ B / (2 A) ] }2 + [ C - B2 / ( 4 A ) ]. The term C - B2 / ( 4 A ) is the y-coordinate of the vertex point, which happens to be - D / ( 4 A ). Therefore if D = 0, the root of the quadratic equation is exactly the vertex. If D > 0 and A > 0, that means the minimum point is below x-axis and the parabola is opened upwards, then there must be two real roots. Similarly, If D > 0 and A < 0, that means the maximum point is above x-axis and the parabola is opened downwards, then there must be two real roots. For D < 0 and A > 0, that means the minimum point is above x-axis and the parabola is opened upwards, then there must not be any real roots. For D < 0 and A < 0, that means the maximum point is below x-axis and the parabola is opened downwards, then there must not be any real roots. What is the quadratic formula? The quadratic formula is a formula to compute the roots of A x2 + B x + C = 0. The roots of the quadratic equations can be founded by −B ± B 2 − 4AC x= 2A What are the properties of product of the roots and the sum of the roots of a quadratic equation? The product of the roots is C / A and the sum of the roots is - B / A. To see these, if we represent a quadratic equation with its roots α1, α2 in this way: A(x − α1 )(x − α 2 ) = 0 After expansion, Ax 2 − A(α1 + α 2 ) + A(α1α 2 ) = 0 By comparing the coefficient with A x2 + B x + C = 0, we get B = - A * (sum of roots)"" " and C = A * (product of roots) which yields sum of roots = - B / A" " " and product of roots = C / A