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A quadratic function is a polynomial function in a single variable, x, of the form equivalent to f(x)=ax2+bx+c with a≠0. The expression of a quadratic function is therefore called a polynimial of degree 2, or the second order or second degree polynomial with the highest exponent of the single variable, x equal to 2. And a quadratic equation is any equation of the form equivalent to ax2+bx+c=0 by setting the quadratic function equal to zero. Therefore a quadratic equation is only a typical example of a quadratic function.
Roots of Quadratic Equations
The solutions of a quadratic equation is also called roots of a quadratic equation. Since a quadratic equation is a second order polynomial equation, a quadratic equation must has two solutions according to the fundamental theorem of algebra. In general, the solution may be both real, or both complex. Assume α and β are the roots of the quadratic equation, ax2+bx+c=0, the quadratic equation can then be written as (x-α)(x-β)=ax2-(α+β)x+αβ=0, imply (α+β)=-b/a and αβ=c/a by equating the coefficients of two polynomials.
Quadratic Formula of Quadratic Equations
Analytically, the solutions of quadratic equation can be found by the method of "completing the square". The idea of determining the roots of a quadratic equation is making use of the symmetry of the quadratic function g(x) about a vertical axis parallel to the y-axis by transforming the quadratic equation f(x)=0 into a symmetry quadratic function g(x). The obtained formula for calculating the roots of the quadratic equation is called the quadratic formula. Imply
Assuming α≥β then the values of the α and β are
Discriminant of Quadratic Equations
According to the quadratic formula, the roots of a quadratic equation is charactered by the square root components of the quadratic formula. Depending on the value of expression (b2-4ac), the roots of a quadratic equation can be coincident, rational, real or complex numbers. And (b2-4ac) is therefore called the discriminant of the quadratic equation. In general, the discriminant, b2-4ac, can have four cases:
Alternate Form of Quadratic Formula
The quadratic equation is obtained by setting the quadratic function equal to zero and x is not equal to zero, the quadratic formula can be derived in an alternative way.
Forms of Quadratic Function
The quadratic function f(x)=ax2+bx+c is usually called the standard form of a quadratic function. Since a quadratic equation has two roots, the standard form of a quadratic function can be expressed in terms of the two roots, f(x)=a(x-x1)(x-x2), called factored form.
Besides the quadratic function is symmetric about a line, the standard form of a quadratic function can be expressed in terms of the vertex of the function, f(x)=a(x-x0)2+y0, called vertex form.
Sign of Quadratic Expression
Let E be the quadratic expression of the quadratic function f(x)=ax2+bx+c. If α and β are the roots of quadratic equation ax2+bx+c=0 and α≥β, then the sign of the quadratic expression can also be determined accordingly. In general, assuming the quadratic expression with two distinct real roots, when a is positive, E is positive if x>α or x<β and E is negative if α>x>β. Or when a is negative, E is positive if α>x>β and E is negative if x>α or x<β. In other words, E has same sign as coefficient a for x>α or x<β, and E has opposite sign as coefficient a for α>x>β. This case can also be considered as the case b2-4ac>0 with two distinct real root for the corresponding quadratic equation. Imply
In the case b2-4ac=0 with double real root for the corresponding quadratic equation, this is a special condition of the general case where α equal to β and therefore the condition α>x>β becomes α=x=β is equal to zero. E has the same sign as the coefficient a.
In the case b2-4ac<0 with two distinct complex root for the corresponding quadratic equation, this is a special condition of the general case where quadratic expression, E, does not equal to zero for all real x. E has the same sign as the coefficient a..
ID: 130500017 Last Updated: 2013/5/17 Revision: Ref:
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