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Here are a few recommended readings before getting started with this lesson.
Paulina's birthday is this weekend and her parents have hidden her gifts in a trunk. She can have them early if she can open the combination lock on the truck. Her parents gave her the clues to help her find the combination, and she has figured out all but the final two digits.
Paulina knows that the last digit is 2 units greater than the one before it. She also knows that the second to last digit is a solution to the following quadratic equation.Split into factors
Commutative Property of Multiplication
a2+2ab+b2=(a+b)2
In the following applet, use the method of completing the square to determine the value of c that makes the given expression a perfect square trinomial. Round to 2 decimal places if needed.
The most useful application of completing the square is that it can be extended to solve quadratic equations. However, some additional steps need to be taken when using this method to solve equations.
Split into factors
Factor out 2
LHS/2=RHS/2
LHS−2=RHS−2
Split into factors
a2+2ab+b2=(a+b)2
Finally, the resulting equations of the previous step need to be solved. These solutions will also be solutions to the original equation.
x+3=±7 | ||
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Write as two equations | x+3=7 | x+3=-7 |
Solve for x | x=-3+7 | x=-3−7 |
Therefore, the solutions of the given equation are x=-3+7 and x=-3−7.
While Paulina thinks about finding the missing digits of the combination lock, her older brother, Vincenzo, and her parents are setting up a rectangular pool for her birthday party. The pool will be in the backyard and will cover an area of 768 square feet. Additionally, they want the length of the pool to be 32 feet longer than the width.
Answer the following questions to help Vincenzo and his parents find the dimensions they should use for the pool.
Split into factors
a2+2ab+b2=(a+b)2
Calculate power
Add terms
x+16=±32 | |
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x+16=32 | x+16=-32 |
x=16 | x=-48 |
At Paulina's birthday party, there will be a lemonade dispenser that automatically fills people's glasses. The dispenser has a capacity of 40 liters and it is expected to be emptied after 60 minutes. The dispenser must be refilled when there is only one liter of lemonade left in it in order for the automatic filling function to work.
The following quadratic equation expresses the volume V of liquid in the dispenser after t minutes.Substitute 1 into the equation for V and solve it by completing the square.
LHS+552=RHS+552
Split into factors
a2−2ab+b2=(a−b)2
LHS=RHS
Calculate root
a2=a
LHS+55=RHS+55
Dominika and Heichi built a small rocket for Paulina's birthday party. They all excitedly decide to launch it at the end of her birthday party.
The rocket has an initial vertical velocity of 32 feet per second. Additionally, the rocket will be launched from a height of 12 feet above the ground. The following quadratic equation describes the height of the rocket, where t is the time in seconds.No, see solution.
The square of a real number cannot be negative.
LHS−12=RHS−12
LHS⋅(-1)=RHS⋅(-1)
Distribute (-1)
(-a)(-b)=a⋅b
a(-b)=-a⋅b
a+(-b)=a−b
Split into factors
Factor out 16
LHS/16=RHS/16
Put minus sign in front of fraction
A⋅Aa=a
LHS+1=RHS+1
a2−2ab+b2=(a−b)2
Add terms