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).