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Quadratic Polynomials

The Quadratic Formula

The method of completing the square can be applied to any quadratic polynomial.

You simply rewrite      ax2+bx+c = a(x2+b over ax)+c

From it we can obtain the following result:

The roots of   ax2+bx+c   are given by

x = (-b + or minus sqrt(b^2 - 4*a*c))/(2*a) (Quadratic Formula)

The quantity   b2−4ac   is called the discriminant of the polynomial.

  • If   b2−4ac < 0   the equation has no real number solutions, but it does have complex solutions.
  • If   b2−4ac = 0   the equation has a repeated real number root.
  • If   b2−4ac > 0   the equation has two distinct real number roots.

Example

Study some of these examples:

Find the roots of    x2 + x + = 0

 

x =
 ± sqrt( 2 − 4× × )


x =
 ± sqrt( )


x =
 , 

 

Example

  x2 + x +

b2 - 4ac =

roots

x1 = , x2 =

Exercise

Now try some of these exercises:

The roots of    x2 + x +    are:

Working area: 

                     

Parabola Vertex

Note that if the roots of a quadratic equation  ax2+bx+c  are real and distinct, then the vertex of the parabola given by the polynomial is situated where

x=-b/(2*a)

 

Example

Study a few of these examples:

Locating the vertex of the parabola given by    x2 + x +  : 

The x-coordinate is

x =
  =

Substituting this value of x into the given equation we find:

the y-coordinate is    ( )2 + ( ) + =

Hence the vertex is ( , )

Exercise

Now try some of these exercises. Give your answers rounded to 2 decimal places:

Locate the vertex of the parabola given by    x2 + x +  : 

Working area: 

The vertex is ( , )

If the roots of a quadratic equation   ax2+bx+c   are α and β, then we can write   ax2+bx+c = a(x−α)(x−β)

 

<< Completing the Square | Quadratic Polynomials Index | Quadratic Functions Factoriser >>

 

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