Quadratic formula, maths first, institute of fundamental sciences, massey university

     

The Quadratic Formula

The method of completing the square can be applied to 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

*
(Quadratic Formula)

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:




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Find the roots of x2 + x + = 0


x =
± sqrt( 2 − 4× × )

x =
± sqrt( )

x =
,

Example




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x2 + x +


b2 - 4ac =

roots

x1 = , x2 =

Exercise

Now try some of these exercises:

The roots of x2 + x + are:

Working area:

Parabola Vertex

cảnh báo that if the roots of a quadratic equationax2+bx+c are real & 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 =

=


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 sầu 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 & 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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