How can you find the time it takes an accelerating cart to reach the finish line?

  

WritinginMathUsetheinformationonpage 288toexplainhowyoucanfindthetimeittakesanacceleratingcartoreachthefinishline.Includeanexplanationofwhyt^2+22t+121=246cannotbesolvedbyfactoringandadescriptionofthestepsyouwouldtaketosolvetheequation.
5-5
Completing the Square
Main Ideas
. Solve quadratic
equations by using
the Square Root
Property
. Solve quadratic
equations by
completing the
square
GET READY for the Lesson
Under a yellow caution flag, race car
drivers slow to a speed of 60 miles per
hour. When the green flag is waved,
the drivers can increase their speed.
Suppose the driver of one car is 500
feet from the finish line. If the driver
accelerates at a constant rate of 8 feet
per second squared, the equation
12 +22+ + 121 = 246 represents the
time t it takes the driver to reach this
line. To solve this equation, you can use
the Square Root Property.
New Vocabulary
completing the square
Square Root Property You have solved equations like x2 25 = 0 by
factoring. You can also use the Square Root Property to solve such an
equation. This method is useful with equations like the one above that
describes the race car’s speed. In this case, the quadratic equation
contains a perfect square trinomial set equal to a constant.
EXAMPLE Equation with Rational Roots
Solve x? + 10x + 25 = 49 by using the Square Root Property.
x? + 10x + 25 = 49
Original equation
(x + 5)2 = 49
Factor the perfect square trinomial.
x+5=149 Square Root Property
*+5 = +7
V 497
= 2
I= -12
x= -5 +7 Add-5 to each side.
x=-5+7 or r=-5-7 Write as two equations.
Solve each equation
The solution set is (2, -12). You can check this result by using factoring
to solve the original equation.
CHECK Your Progress
Solve each equation by using the Square Root Property.
1A. x2 – 12x + 36 = 25
1B. x2 – 16x + 64 = 49
Roots that are irrational numbers may be written as exact answers in radical
form or as approximate answers in decimal form when a calculator is used.
268 Chapter 5 Quadratic Functions and Inequalities

Introduction:
Mathematics is a subject that demands a clear understanding of relevant concepts and logical reasoning. The field of math comprises numerous equations and methods which help in solving the problems. When it comes to quadratic equations, there are various techniques to solve them effectively. In this article, we will discuss how to use the Square Root Property and completing the square methods for solving quadratic equations.

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Description:
One of the key techniques used to solve quadratic equations is the Square Root Property. When a quadratic equation consists of a perfect square trinomial set equal to a constant, one can use the Square Root Property to solve it. Completing the square is another useful method for solving quadratic equations. To understand the application of these methods, let’s take an example of a race car driver. Suppose a driver is 500 feet away from the finishing line and accelerates at a constant rate of 8 feet per second squared. In this case, we need to find the time taken by the driver to reach the finishing line. The equation to find the time is t^2+22t+121=246. However, this equation cannot be solved by factoring. Therefore, we need to use the completing the square method to determine the value of t. By using the Square Root Property, we can solve quadratic equations that contain a perfect square trinomial. One can also represent the irrational roots in decimal form through a calculator. In summary, the Square Root Property and completing the square methods are helpful techniques to solve quadratic equations.

Objectives:
1. Understand how completing the square method is used to solve quadratic equations.
2. Learn to use the Square Root Property to solve quadratic equations.
3. Understand how to apply the completing the square method and Square Root Property to solve equations in real-world scenarios.

Learning Outcomes:
By the end of this lesson, students will be able to:
1. Solve quadratic equations using the completing the square method.
2. Use the Square Root Property to solve quadratic equations.
3. Apply the completing the square method and Square Root Property to solve equations related to real-world scenarios, such as finding the time it takes for an accelerating car to reach the finish line.
4. Demonstrate the ability to check solutions obtained using the completing the square method and Square Root Property by using factoring.

Solution 1: Completing the Square

To find the time it takes for the accelerating car to reach the finish line, we need to solve the quadratic equation t^2 + 22t + 121 = 246. However, this equation cannot be solved by factoring. To solve it, we can use completing the square method.

First, we need to move the constant over to the right side of the equation, so we have t^2 + 22t = 125.

Next, we need to calculate half of the coefficient of t and square it, which gives us (22/2)^2 = 121. We add 121 to both sides of the equation to complete the square, so we have t^2 + 22t + 121 = 246.

Now, we can factor the left side of the equation into a perfect square: (t + 11)^2 = 246.

Finally, we take the square root of both sides of the equation and solve for t: t + 11 = ±√246. Subtracting 11 from both sides, we get two solutions: t = -11 ± √246.

Therefore, the time it takes for the accelerating car to reach the finish line is approximately 6.26 seconds or -28.26 seconds, but the negative solution does not make sense in the context of the problem.

Solution 2: Square Root Property

Another way to solve the quadratic equation t^2 + 22t + 121 = 246 is to use the Square Root Property.

First, we need to move the constant over to the right side of the equation, so we have t^2 + 22t = 125.

Next, we divide both sides of the equation by the coefficient of t^2, which is 1, so we have t^2 + 22t/1 = 125/1.

Now, we add (22/2)^2 = 121 to both sides of the equation to complete the square, so we have t^2 + 22t + 121 = 246.

Taking the square root of both sides of the equation, we get t + 11 = ±√246. Subtracting 11 from both sides, we get two solutions: t = -11 ± √246.

Therefore, the time it takes for the accelerating car to reach the finish line is approximately 6.26 seconds or -28.26 seconds, but the negative solution does not make sense in the context of the problem.

Suggested Resources/Books:

1. “Algebra and Trigonometry” by Stewart, Redlin and Watson
2. “Precalculus: Mathematics for Calculus” by Stewart, Redlin and Watson
3. “Calculus: Early Transcendentals” by Stewart
4. “Mathematics: Its Power and Utility” by Karl Smith

Similar Asked Questions:

1. How do you solve quadratic equations by completing the square?
2. What is the Square Root Property and how is it used to solve equations?
3. How can factoring be used to solve quadratic equations?
4. How do you find the vertex of a parabola given a quadratic equation?
5. How can quadratic equations be applied to real-world situations, like the example given with the accelerating car?

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