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+x)+c

From it we can obtain the following result:

The roots of   ax²+bx+c   are given by

(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× × ) 2× 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

 

Example

Study a few of these examples:

Locating the vertex of the parabola given by    x2 + x +  :  The x-coordinate is x =   = 2× 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)

 

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