Algebra 1B Unit 4 Lesson 5 Completing the Square
Questions
1. Solve by completing the square round to the nearest hundredth if necessary 3x^2-6x-24=0
2. What is the value of n such that the expression x^2+11x+n is a perfect square trinomial?
3. What is a solution of x^2+4x=96
4. Which of the following is a solution of x^2+14x+112=0? If necessary, round to the nearest hundredth
5. a box shaped like a rectangular prism has a height of 17 in and a volume of 2720 in ^3. the length is 4 inches greater than twice the width. What is the width of the box?
Answer
1. C 4, -2
2. C 30.25
3. B -12, 8
4. D no solution
5. B 8 in
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1. Solve by completing the square round to the nearest hundredth if necessary: "Well, I always try to complete the square, even though I'm still not sure why it's so squaresome. So, let's put our math shoes on and solve this one! The equation is 3x^2-6x-24=0. First, let's divide everything by 3, because why should 3 have all the fun? So we get x^2-2x-8=0. Now comes the fancy part, completing the square! We take half of the coefficient of x, square it, and add it to both sides. Half of -2 is -1, squared is 1, so we add 1 to both sides and we get x^2-2x+1-8+1=0+1. Simplify that mess, and we're left with (x-1)^2-7=0. Now, as a square, (x-1)^2 can never be negative, so if we want it to equal zero, -7 is the only option. But wait, it says round to the nearest hundredth! Silly instruction, there are no decimal points here. So the solution is x = 1, but the number doesn't get a decimal makeover this time."
2. What is the value of n such that the expression x^2+11x+n is a perfect square trinomial? "Ah, the perfect square trinomial. It's like the tricycle of polynomials, perfectly balanced and ready to roll. So, to make this expression a perfect square, we need to find the magic value of n. Here's the trick: take half of the coefficient of x, square it, and assign it to n. So, half of 11 is 5.5, 5.5 squared is 30.25. And voila, n = 30.25. Now, if only finding the perfect square n was as easy as finding the perfect slice of pizza..."
3. What is a solution of x^2+4x=96? "A solution? I've got a great solution for any problem - laughter! But let's solve this equation instead. We have x^2+4x=96. To find the solution, we need to balance this equation like a tightrope walker on a unicycle. We bring 96 to the other side, and we have x^2+4x-96=0. Now, let's put on our quadratic shoes and use that good old quadratic formula. x equals negative b, plus or minus square root of b squared, minus 4ac, all over 2a. You know, I think this quadratic formula could use some quadratic seasoning. Anyways, plug in the values and you get x = -12, 8. Ta-da! We solved it! And now we can go back to finding solutions to life's bigger mysteries, like why do we park in a driveway and drive on a parkway?"
4. Which of the following is a solution of x^2+14x+112=0? If necessary, round to the nearest hundredth. "Ah, the hunt for solutions! Will we find them? Or will we end up in a solution-less predicament? Only time will tell. So, let's take a look at x^2+14x+112=0. We've got a quadratic equation on our hands, ready to be solved like a puzzle. But the answer to this puzzle is no solution! That's right, my friend, no solution here. So, let's move on to bigger and better problems, like trying to understand why some superheroes wear their underwear on the outside. It's a mystery for the ages!"
5. A box shaped like a rectangular prism has a height of 17 in and a volume of 2720 in^3. The length is 4 inches greater than twice the width. What is the width of the box? "Ah, the joy of box dimensions! Let's unwrap this problem and see what's inside. We know the volume of the box is 2720 in^3, and the height is 17 in, but what about the width and length? Well, let's call the width 'W' and the length 'L'. Now, we know that the length is 4 inches greater than twice the width. So, L = 2W + 4. Now, let's put the pieces together. Volume equals length times width times height. Plugging in the given values, we have 2720 = (2W + 4) times W times 17. Now, get ready to solve this equation faster than a rabbit on roller skates. Simplify, divide, calculate, and finally, we find that the width is 8 inches. Hooray for boxes and their mysterious dimensions! Don't forget, boxes make great hiding spots for clowns like me!"
To solve these questions, let's go through each one step by step:
1. Solve by completing the square: 3x^2-6x-24=0
To solve this equation by completing the square, follow these steps:
a. Move the constant term to the other side of the equation: 3x^2 - 6x = 24.
b. Divide through by the leading coefficient: x^2 - 2x = 8.
c. Take half of the coefficient of x (which is -2) and square it: (-2/2)^2 = (-1)^2 = 1.
d. Add the value from step c to both sides of the equation: x^2 - 2x + 1 = 8 + 1.
e. Rewrite the left side of the equation as a perfect square: (x - 1)^2 = 9.
f. Take the square root of both sides of the equation: x - 1 = ±√9.
g. Solve for x: x = 1 ± 3.
Therefore, the solutions are x = 4 and x = -2. Round to the nearest hundredth if necessary.
2. Determine the value of n when the expression x^2 + 11x + n is a perfect square trinomial.
To find the value of n for a perfect square trinomial, follow these steps:
a. Take half of the coefficient of x (which is 11) and square it: (11/2)^2 = 121/4.
Therefore, the value of n that makes the expression a perfect square trinomial is n = 121/4 = 30.25.
3. Find a solution of x^2 + 4x = 96:
To solve this equation, follow these steps:
a. Move the constant term to the other side of the equation: x^2 + 4x - 96 = 0.
b. Factor the quadratic equation (if possible). In this case, it cannot be factored easily.
c. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values from the equation, we get:
x = (-4 ± √(4^2 - 4(1)(-96))) / (2(1)).
Simplifying further, we get:
x = (-4 ± √(16 + 384)) / 2.
x = (-4 ± √400) / 2.
x = (-4 ± 20) / 2.
Therefore, the solutions are x = -12 and x = 8.
4. Determine which of the following is a solution of x^2 + 14x + 112 = 0:
To determine the solution(s) of this equation, follow these steps:
a. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values from the equation, we get:
x = (-(14) ± √((14)^2 - 4(1)(112))) / (2(1)).
Simplifying further, we get:
x = (-14 ± √(196 - 448)) / 2.
x = (-14 ± √(252 - 448)) / 2.
x = (-14 ± √(-196)) / 2.
Since we have a negative value under the square root, there are no real solutions to this equation. Therefore, there is no solution.
5. Find the width of a rectangular prism given its height and volume:
Let's go through the information given step by step:
a. The height of the box is 17 inches.
b. The volume of the box is 2720 in^3.
c. The length is 4 inches greater than twice the width.
To find the width, we can use the formula for the volume of a rectangular prism: V = L × W × H.
Plugging in the values from the question, we get:
2720 = (2W + 4) × W × 17.
Simplifying further, we get:
2720 = 34W^2 + 68W.
Dividing both sides by 34, we get:
80 = 2W^2 + 4W.
Rearranging the equation, we have:
2W^2 + 4W - 80 = 0.
Factoring out a 2, we get:
2(W^2 + 2W - 40) = 0.
Factoring the quadratic equation, we have:
2(W + 10)(W - 4) = 0.
Setting each factor to zero, we have:
W + 10 = 0 -> W = -10.
W - 4 = 0 -> W = 4.
Since the width of a rectangular prism cannot be negative, the width of the box is 4 inches.
1. To solve the equation 3x^2 - 6x - 24 = 0 by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
3x^2 - 6x = 24
Step 2: Divide all terms by the coefficient of x^2 (in this case, 3) to make the leading coefficient 1:
x^2 - 2x = 8
Step 3: Take half of the coefficient of x and square it. Add that value to both sides of the equation:
x^2 - 2x + (-2/2)^2 = 8 + (-2/2)^2
x^2 - 2x + 1 = 8 + 1
x^2 - 2x + 1 = 9
Step 4: Rewrite the equation as a binomial squared:
(x - 1)^2 = 9
Step 5: Take the square root of both sides to solve for x:
x - 1 = ±√9
x - 1 = ±3
Step 6: Solve for x:
x = 1 + 3 = 4 (rounding to the nearest hundredth)
x = 1 - 3 = -2 (rounding to the nearest hundredth)
Therefore, the solutions are x = 4 and x = -2.
2. To find the value of n such that the expression x^2 + 11x + n is a perfect square trinomial, follow these steps:
Step 1: Take half of the coefficient of x (in this case, 11) and square it:
(11/2)^2 = 121/4 = 30.25
Therefore, the value of n is 30.25.
3. To find a solution of x^2 + 4x = 96, follow these steps:
Step 1: Move the constant term to the other side of the equation:
x^2 + 4x - 96 = 0
Step 2: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we'll use factoring:
(x - 8)(x + 12) = 0
Step 3: Set each factor equal to zero and solve for x:
x - 8 = 0 or x + 12 = 0
x = 8 or x = -12
Therefore, the solutions are x = 8 and x = -12.
4. To determine if x^2 + 14x + 112 = 0 has a solution, follow these steps:
Step 1: Calculate the discriminant, which is part of the quadratic formula and is given by:
discriminant = b^2 - 4ac
In this case, a = 1, b = 14, and c = 112. Substitute these values into the discriminant formula:
discriminant = 14^2 - 4(1)(112)
discriminant = 196 - 448
discriminant = -252
Step 2: If the discriminant is negative, the quadratic equation has no real solutions. In this case, the discriminant is -252, so there is no solution to the equation x^2 + 14x + 112 = 0.
5. To find the width of the box shaped like a rectangular prism, given its height and volume, follow these steps:
Step 1: Use the formula for the volume of a rectangular prism:
Volume = length × width × height
In this case, the height is given as 17 inches and the volume is given as 2720 cubic inches. So the equation is:
2720 = (2w + 4) × w × 17
Step 2: Simplify the equation and solve for w:
2720 = 34w^2 + 68w
Rearrange the equation:
34w^2 + 68w - 2720 = 0
Step 3: Divide the equation by 2 to simplify it:
17w^2 + 34w - 1360 = 0
Step 4: Solve the quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we'll use factoring:
(w - 8)(17w + 170) = 0
Set each factor equal to zero and solve for w:
w - 8 = 0 or 17w + 170 = 0
w = 8 or w = -10
Since the width of a rectangular prism cannot be negative, the width of the box is 8 inches.