The method of completing the square can be applied lớn 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 The quantity b2−4ac is called the discriminant of the polynomial.

If b2−4ac 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 = ,

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 equationax2+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 khổng lồ 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 và 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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