Complete the square in the quadratic function f given by
f(x) = 2x2 - 6x + 4

Start with the given function:

f

(

x

)

=

2

x

2

−

6

x

+

4

Make the leading coefficient

1

by factoring

2

out of the first two terms:

f

(

x

)

=

2

(

x

2

−

3

x

)

+

4

Consider the binomial

x

2

−

3

x

. To complete the square, take the coefficient of the linear term,

−

3

, and square its half:

9

4

. This constant must be added to the binomial to complete the square. Note that by doing so, it changes the value entirely, and so the change must be undone. Note that adding this constant to the binomial effectively adds twice its value because of the factored

2

. Thus, to undo the change brought by the addition, subtract the entire expression by twice the constant:

9

2

. This gives the following:

f

(

x

)

=

2

(

x

2

−

3

x

+

9

4

)

+

4

−

9

2

The expression is now a perfect square trinomial. Simplifying the outside constants and expressing the trinomial as a perfect square, the following is written:

f

(

x

)

=

2

(

x

−

3

2

)

2

−

1

2

This rounds up the process of completing the square.

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