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| 9 Theory slides |
| 9 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
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.
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.
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
x+1=±5 | |
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x+1=5 | x+1=-5 |
x=4 | x=-6 |
There are two solutions for the equation. However, since the combination lock does not include negative numbers, only 4 makes sense. Therefore, the second to last number of the code is 4.
A right triangle has a hypotenuse of 25 centimeters. One of the legs is 8 centimeters longer than the other leg. Determine the area of the triangle. Round to two decimal places.
We will use the Pythagorean Theorem to write a quadratic equation for the side lengths of the triangle. Solving this equation will allow us to find the missing side lengths of the triangle.
Consider the Pythagorean Theorem. a^2+b^2=c^2 We will substitute 25 for c, x for a, and x+8 for b into the Pythagorean Theorem.
The quadratic equation obtained previously can be solved by completing the square. To do so, let's first rewrite it in the form x^2+bx=c.
To produce a perfect square trinomial on the left-hand side of the equation, we must calculate ( b2)^2. This term should be later added to both sides of the equation. In this case, b= 8.
We will now add this term to both sides of the equation and factor the resulting perfect square trinomial.
Let's apply the square root to both sides of the equation to find its solutions.
Let's analyze each case and use a calculator to find the solutions to the equation.
x=-4±sqrt(593/2) | |
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x=-4+sqrt(593/2) | x=-4-sqrt(593/2) |
x≈ 13.22 | x≈ -21.22 |
We have found the solutions to the equation. However, because x represents the measure of one of the legs of the triangle, x cannot be negative. This means we need to consider only the positive solution x≈ 13.22.
We found that x≈ 13.22. This means the other leg of the triangle is 13.22+8≈ 21.22 centimeters. Consider the formula for the area of a triangle. A=1/2b* h We will substitute 13.22 for h and 21.22 for b to find the area of the triangle.
Therefore, the area of the triangle is about 140.26 square centimeters.
Find the area of the rectangle and round it to one decimal place.
The side lengths are needed to determine the area of the rectangle. Since the diagonal forms a right triangle with the base and height of the rectangle, we can use the Pythagorean Theorem to determine the value of x.
Note that we have written the equation in the form x^2+bx=c. This means we can use the method of completing the square to find its solutions. We will do this by first calculating the missing term ( b2)^2, using b= 0.8.
Adding this term to both sides of the equation will produce a perfect square trinomial on its left-hand side.
We can now apply the square root to both sides of the equation to obtain its solutions.
Let's analyze each case and use a calculator to find the solutions to the equations.
x=-0.4±sqrt(9.76) | |
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x=-0.4+sqrt(9.76) | x=-0.4-sqrt(9.76) |
x≈ 2.7 | x≈ -3.5 |
Since side lengths cannot be negative, we need to consider only the positive solution x=2.7. Let's now consider the formula for the area of a rectangle. A= l w We will substitute 2x-3 for l and x+8 for w to obtain an expression for the area of the given rectangle. A= l w Substitute A=( 2x-3)( x+8) Let's evaluate this expression when x ≈ 2.7 to obtain the area of the rectangle.
We have found that the area of the rectangle is about 25.7 square units.