 Home > College of Sciences > Institute of Fundamental Sciences > Maths First > Online Maths Help > Algebra > Quadratic Polynomials > Quadratic Formula SEARCH MASSEY  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)