If f(x) = x2 − 6x + 19, complete the square and determine the minimum or maximum value of the function

minimum value = 10

Step-by-step explanation:

The equation of a parabola in vertex form is

y = a(x - h)² + k

where (h, k) are the coordinates of the vertex and a is a multiplier

To obtain this form using completing the square

add/subtract ( half the coefficient of the x- term )² to x² - 6x

f(x) = x² + 2(- 3)x + 9 - 9 + 19

     = (x - 3)² + 10 ← in vertex form

with (h, k) = (3, 10 )

To determine maximum or minimum consider the value of a

• If a > 0 then minimum value of k

• If a < 0 then maximum value of k

Here a = 1 > 0, hence minimum value of k = 10

answer is 8x+24

area of vanessa's patio = length × width

given, length = (3x² + 5x + 10)

width = (x² - 3x - 1)

area of vanessa's patio = (3x² + 5x + 10)(x² - 3x - 1)

= 3x²(x² - 3x - 1) + 5x(x² - 3x - 1) + 10(x² - 3x - 1)

= 3x⁴ - 9x³ - 3x² + 5x³ - 15x² - 5x + 10x² - 30x - 10

= 3x⁴ - 4x³ - 8x² - 35x - 10

hence, expression of vanessa's patio = (3x⁴ - 4x³ - 8x² - 35x - 10)

96 39 cause 135 -96=39 im smart (i used a calculater)