Algebra Quadratic Equations Algebra Quadratic Equations In mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the form where x represents a variable or an unknown, and a, b, and c are constants with a ≠ 0. (If a = 0, the equation is a linear equation.) The constants a, b, and c are called respectively, the quadratic coefficient, the linear coefficient and the constant term or free term. The term "quadratic" comes from quadratus, which is the Latin word for "square". Quadratic equations can be solved by factoring, completing the square, graphing, Newton's method, and using the quadratic formula (given below). The polynomial which can be expressed in the form of ax2 + bx + c = 0, then we say that the equation is in the form of quadratic polynomial. Here we say that a, b, c are the real Numbers and we must remember that a <> 0, since if we have a = 0, the equation will convert into a linear equation in place of the quadratic equation.
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If we say that alpha (α) and beta (β) are the two roots of the Quadratic Equation and their sum of the root i.e. α + β is written as = - coefficient of ‘x’ / coefficient of x2. Also the products of the roots is written as α * β = c/a, Now in case the roots of the equation are known, then we can form the quadratic equation using the following formula: x2 – sum of roots * x + product of roots = 0 We can find the solution of the quadratic equations by Factorization, by completing the squares and making them the perfect squares and it is also done even by the quadratic formula. Once we learn to use the formula of the quadratic roots and to find the value of the determinants, and the nature of roots can also be known. Now let us see that α and β are the roots of the quadratic equation, then it means that if we put the value of α or β in the given equation, then it satisfies the given equation. D = b2 – 4 * a * c is the formula which helps to analyze the types of roots of the equation. If D = 0, then roots are real and equal, if D> 0, then roots are unequal and real, if D< 0, then roots are imaginary. Geometric solution :- The quadratic equation may be solved geometrically in a number of ways. One way is via Lill's method. The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in the accompanying diagram. A circle is drawn with the start and end point SC as a diameter.
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If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient a or SA. If a is 1 the coefficients may be read off directly. Thus the solutions in the diagram are −AX1/SA and −AX2/SA. Generalization of quadratic equation :- The formula and its derivation remain correct if the coefficients a, b and c are complex numbers, or more generally members of any field whose characteristic is not 2. (In a field of characteristic 2, the element 2a is zero and it is impossible to divide by it.) In the formula should be understood as "either of the two elements whose square is b2 − 4ac, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. Note that even if a field does not contain a square root of some number, there is always a quadratic extension field which does, so the quadratic formula will always make sense as a formula in that extension field.
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