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Quadratic Equations


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Quadratic Equations

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.

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Roots of Quadratic Equations[1]

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.

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Quadratic Formula of Quadratic Equations[1]

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

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Assuming α≥β then the values of the α and β are

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Discriminant of Quadratic Equations[1]

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:

  1. b2>4ac, imply discriminant, b2-4ac>0, if b2-4ac is a perfect or complete square then two distinct roots are rational.

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  2. b2>4ac, imply discriminant, b2-4ac>0, if b2-4ac is not a perfect or complete square then two distinct roots are real or irrational.

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  3. b2=4ac, imply discriminant, b2-4ac=0, two roots are rational and coincident. That is two equal roots, one double root or one single solution.

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  4. b2<4ac, imply discriminant, b2-4ac<0, two distinct roots are complex or imaginary.

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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.

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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.

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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.

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Sign of Quadratic Expression[1]

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

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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.

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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..

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References

  1. B. Joseph, 1978, University Mathematics: A Textbook for Students of Science & Engineering, Blackie & Son Limited, HongKong
  2. Wheatstone, C., 1854, On the Formation of Powers from Arithmetical Progressions, Proceedings of The Royal Society of London, Vol 7, p145-151,, London
  3. Stroud, K.A., 2001, Engineering Mathematics, Industrial Press, Inc, NY
  4. Coolidge, J.L., 1949, The Story of The Binomial Theorem, The American Mathematical Monthly, Vol 56, No.3, Mar, pp147-157
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ID: 130500017 Last Updated: 2013/5/17 Revision: Ref:

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