Question

The equation 2x2 − 12x + 1 = 0 is being rewritten in vertex form. Fill in the missing step. Given 2x2 − 12x + 1 = 0 Step 1 2(x2 − 6x ___) + 1 ___ = 0 Step 2 2(x2 − 6x + 9) + 1 − 18 = 0 Step 3 ✔ 2(x − 3)2 + 17 = 0 2(x − 3)2 − 17 = 0 2(x + 3)2 − 17 = 0 2(x − 6)2 − 17 = 0 Question 3(Multiple Choice Worth 1 points) (08.02 MC) The function f(x) = −x2 + 16x − 60 models the daily profit, in dollars, a shop makes for selling candles, where x is the number of candles sold. Determine the vertex, and explain what it means in the context of the problem. (6, 10); The vertex represents the maximum profit. (6, 10); The vertex represents the minimum profit. (8, 4); The vertex represents the minimum profit. (8, 4); The vertex represents the maximum profit. Question 4(Multiple Choice Worth 1 points) (08.02 MC) Solve for x: −2(x − 2)2 + 5 = 0 Round your answer to the nearest hundredth. x = 3.58, 0.42 x = 4.52, −0.52 x = −3.58, −0.42 x = −4.52, 0.52 Question 5(Multiple Choice Worth 1 points) (08.02 MC) A yoga studio sells monthly memberships. The function f(x) = −x2 + 50x − 264 models the profit in dollars, where x is the number of memberships sold. Determine the zeros, and explain what these values mean in the context of the problem. x = 6, x = 44; The zeros represent the number of monthly memberships that produces a maximum profit. x = 6, x = 44; The zeros represent the number of monthly memberships where no profit is made. x = 25, x = 361; The zeros represent the number of monthly memberships where no profit is made. x = 25, x = 361; The zeros represent the number of monthly memberships that produces a maximum profit. Question 6(Multiple Choice Worth 1 points) (08.02 MC) The owner of a video game store creates the expression −2x2 + 28x + 20 to represent the store's weekly profit in dollars, where x represents the price of a new video game. Choose the equivalent expression that reveals the video game price that produces the highest weekly profit, and use it to determine that price. −2(x − 7)2 + 118; x = $7 −2(x − 7)2 + 118; x = $118 −2(x2 − 14x) + 20; x = $14 −2(x2 − 14x − 10); x = $10 Question 7(Multiple Choice Worth 1 points) (08.02 MC) The height of a hockey puck that is hit toward a goal is modeled by the function f(x) = −x2 + 8x − 10, where x is the distance from the point of impact. Complete the square to determine the maximum height of the path of the puck. −(x − 4)2 + 26; The maximum height of the puck is 26 feet. −(x − 4)2 + 26; The maximum height of the puck is 4 feet. −(x − 4)2 + 6; The maximum height of the puck is 4 feet. −(x − 4)2 + 6; The maximum height of the puck is 6 feet. Question 8(Multiple Choice Worth 1 points) (08.02 LC) Complete the square to transform the expression x2 + 6x + 5 into the form a(x − h)2 + k. (x + 6)2 + 4 (x + 6)2 − 4 (x + 3)2 − 4 (x + 3)2 + 4 Question 9(Multiple Choice Worth 1 points) (08.02 MC) Which of the following reveals the minimum value for the equation 2x2 − 4x − 2 = 0? 2(x − 1)2 = 4 2(x − 1)2 = −4 2(x − 2)2 = 4 2(x − 2)2 = −4 Question 10(Multiple Choice Worth 1 points) (08.02 MC) The expression 8x2 − 64x + 720 is used to approximate a small town's population in thousands from 1998 to 2018, where x represents the number of years since 1998. Choose the equivalent expression that is most useful for finding the year where the population was at a minimum. 8(x − 4)2 + 592 8(x − 4)2 − 592 8(x2 − 8x + 90) 8(x2 − 8x) + 90

Answer 1

Part 1)
`2(x-3)^(2)-17=0` (the missing steps in the explanation)

Part 3) (8, 4); The vertex represents the maximum profit

Part 4) x = 3.58, 0.42

Part 5) x = 6, x = 44; The zeros represent the number of monthly memberships where no profit is made

Part 6) 2(x − 7)2 + 118; x = $7

Part 7) The maximum height of the puck is 4 feet. −(x − 4)^2 + 6

Part 8) (x + 3)^2 − 4

Part 9) 2(x − 1)^2 = 4

Part 10) 8(x − 4)^2 + 592

Explanation:

Part 1) we have

`2x^(2) -12x+1=0`

Convert to vertex form

step 1

Factor the leading coefficient and complete the square

`2(x^(2) -6x)+1=0`

`2(x^(2) -6x+9)+1-18=0`

step 2

`2(x^(2) -6x+9)+1-18=0`

`2(x^(2) -6x+9)-17=0`

step 3

Rewrite as perfect squares

`2(x-3)^(2)-17=0`

Part 3) we have

`f(x)=-x^(2)+16x-60`

we know that

This is the equation of a vertical parabola open downward

The vertex is a maximum

Convert to vertex form

`f(x)+60=-x^(2)+16x`

Factor the leading coefficient

`f(x)+60=-(x^(2)-16x)`

Complete the squares

`f(x)+60-64=-(x^(2)-16x+64)`

`f(x)-4=-(x^(2)-16x+64)`

Rewrite as perfect squares

`f(x)-4=-(x-8)^(2)`

`f(x)=-(x-8)^(2)+4`

The vertex is the point (8,4)

The vertex represent the maximum profit

Part 4) Solve for x

we have

`-2(x-2)^(2)+5=0`

`-2(x-2)^(2)=-5`

`(x-2)^(2)=2.5`

square root both sides

`(x-2)=(+/-)1.58`

`x=2(+/-)1.58`

`x=2(+)1.58=3.58`

`x=2(-)1.58=0.42`

Part 5) we have

`f(x)=-x^(2)+50x-264`

we know that

The zeros or x-intercepts are the value of x when the value of the function is equal to zero

so

In this context the zeros represent the number of monthly memberships where no profit is made

To find the zeros equate the function to zero

`-x^(2)+50x-264=0`

`-x^(2)+50x=264`

Factor -1 of the leading coefficient

`-(x^(2)-50x)=264`

Complete the squares

`-(x^(2)-50x+625)=264-625`

`-(x^(2)-50x+625)=-361`

`(x^(2)-50x+625)=361`

Rewrite as perfect squares

`(x-25)^(2)=361`

square root both sides

`(x-25)=(+/-)19`

`x=25(+/-)19`

`x=25(+)19=44`

`x=25(-)19=6`

Part 6) we have

`-2x^(2)+28x+20`

This is a vertical parabola open downward

The vertex is a maximum

Convert the equation into vertex form

Factor the leading coefficient

`-2(x^(2)-14x)+20`

Complete the square

`-2(x^(2)-14x+49)+20+98`

`-2(x^(2)-14x+49)+118`

Rewrite as perfect square

`-2(x-7)^(2)+118`

The vertex is the point (7,118)

therefore

The video game price that produces the highest weekly profit is x=$7

Part 7) we have

`f(x)=-x^(2)+8x-10`

Convert to vertex form

`f(x)+10=-x^(2)+8x`

Factor -1 the leading coefficient

`f(x)+10=-(x^(2)-8x)`

Complete the square

`f(x)+10-16=-(x^(2)-8x+16)`

`f(x)-6=-(x^(2)-8x+16)`

Rewrite as perfect square

`f(x)-6=-(x-4)^(2)`

`f(x)=-(x-4)^(2)+6`

The vertex is the point (4,6)

therefore

The maximum height of the puck is 4 feet.

Part 8) we have

`x^(2)+6x+5`

Convert to vertex form

Group terms

`(x^(2)+6x)+5`

Complete the square

`(x^(2)+6x+9)+5-9`

`(x^(2)+6x+9)-4`

Rewrite as perfect squares

`(x+3)^(2)-4`

Part 9) we have

`2x^(2)-4x-2=0`

This is the equation of a vertical parabola open upward

The vertex is a minimum

Convert to vertex form

Factor 2 the leading coefficient

`2(x^(2)-2x)-2=0`

Complete the square

`2(x^(2)-2x+1)-2-2=0`

`2(x^(2)-2x+1)-4=0`

Rewrite as perfect squares

`2(x-1)^(2)-4=0`

`2(x-1)^(2)=4`

The vertex is the point (1,-4)

Part 10) we have

`8x^(2)-64x+720`

This is the equation of a vertical parabola open upward

The vertex is a minimum

Convert to vertex form

Factor 8 the leading coefficient

`8(x^(2)-8x)+720`

Complete the square

`8(x^(2)-8x+16)+720-128`

`8(x^(2)-8x+16)+592`

Rewrite as perfect squares

`8(x-4)^(2)+592`

the vertex is the point (4,592)

The population has a minimum at x=4 years ( that is after 4 years since 1998 )