complete the square to write each function in the form f(x)=a(x-h)^2+k

f(x)=2x^2-20x-3

2 x ^ 2 - 20 x - 3

2 ( x ^ 2 - 10 x - 3 / 2 )

2 ( x ^ 2 - 10 x + ? - 3 / 2 ) + ?

The quantity to add is "the square of: (1/2) times the coefficient of the middle term -10"

That is:

[ ( 1 / 2 ) * ( - 10 ) ] ^ 2

( - 5 ) ^ 2

2 ( x ^ 2 - 10 x - 3 / 2 )=

2 [ x ^ 2 - 10 x + ( - 5 ) ^ 2 - 3 / 2 ) - ( - 5 ) ^ 2 ]

2 [ x ^ 2 - 10 x + ( - 5 ) ^ 2 ] - 2 * 3 / 2 - 2 * 25

x ^ 2 - 10 x + ( - 5 ) ^ 2 = ( x - 5 ) ^ 2

2 * 3 / 2 = 3

2 * 25 = 50

2 [ x ^ 2 - 10 x + ( - 5 ) ^ 2 - 3 / 2 ) - ( - 5 ) ^ 2 ] =

2 ( x - 5 ) ^ 2 - 3 - 50 =

2 ( x - 5 ) ^ 2 - 53

2 x ^ 2 - 20 x - 3 = 2 ( x - 5 ) ^ 2 - 53

To complete the square and write the function in the form f(x) = a(x-h)^2 + k, follow these steps:

Step 1: Group the x^2 and x terms together.
f(x) = 2x^2 - 20x - 3

Step 2: Factor out the coefficient of x^2 (2 in this case) from the first two terms.
f(x) = 2(x^2 - 10x) - 3

Step 3: To complete the square, take half of the coefficient of x (-10/2 = -5) and square it (-5^2 = 25). Add and subtract this value inside the parentheses.
f(x) = 2(x^2 - 10x + 25 - 25) - 3

Step 4: Rewrite the expression inside the parentheses as a perfect square trinomial.
f(x) = 2((x - 5)^2 - 25) - 3

Step 5: Distribute the 2 to each term inside the parentheses.
f(x) = 2(x - 5)^2 - 50 - 3

Step 6: Combine like terms.
f(x) = 2(x - 5)^2 - 53

So, the function f(x) = 2x^2 - 20x - 3 can be written in the form f(x) = 2(x - 5)^2 - 53. In this form, the vertex of the parabola is at the point (5, -53).