Extraction of Roots

Section 3.1 Extraction of Roots

This is the first Lesson on quadratic equations. Note that we start with extraction of roots, rather than the more traditional solution by factoring.

We recommend going over these Activities with the class, using a smart board or a projector. Almost all of the skills required will be new to most students. During class, take time to stress some of the basic ideas:

How to recognize a quadratic equation, and to obtain the graph of the basic parabola.
Solving an equation by graphing (in particular a quadratic equation).
The difference between an exact solution and a decimal approximation.

Point out that our first quadratic models include the Pythagorean theorem, formulas for volume and surface area, and the compound interest formula.

Activity 3.1.1. Volume.

You are designing a new coffee maker. The coffee filter must be 8.4 centimeters tall and shaped like a cone. The volume of the filter will therefore depend on how wide the opening is (the radius of the cone).

Write a formula for the filter’s volume $V$ in terms of its widest radius $r$ (at the top of the filter). (Hint: Look up the formula for the volume of a cone.)

Simplify your formula by multiplying the constants together.

Complete the table of values. Round the volumes to one decimal place.

$r$ (cm)	1	2	3	4	5	6	7	8
$V$ (cm$^3$)	$\hphantom{000000}$	$\hphantom{000000}$	$\hphantom{000000}$	$\hphantom{000000}$	$\hphantom{000000}$	$\hphantom{000000}$	$\hphantom{000000}$	$\hphantom{000000}$

Look at the table of values. If you double the radius of the filter, by what factor does the volume increase?
Write and solve an equation to answer the question: If the volume of the filter is 302.4 cubic centimeters, what is its radius?
Use technology to graph the volume equation. (Use the table of values to help you choose a window.) Sketch your graph on paper, including scales on the axes). Label the coordinates of the point on the graph that corresponds to the filter in part (d).

Activity 3.1.2. Extraction of Roots.

Solve each equation by extraction of roots.
1. $\displaystyle (x+3)^2=4$
  
  Even students who recently completed elementary algebra have trouble with extraction of roots: although solving $x^2=16$ is easy for them, solving $(x+3)^2=16$ is not.
2. $\displaystyle (3x+1)^2=25$
Solve. Give your answers two ways: (i) exact values and (ii) approximations rounded to thousandths.
1. $\displaystyle 6(x-3)^2=30$
2. $\displaystyle 520+60(2+x)^2=1000$
Solve each formula for the requested variable.
1. $V=\frac{1}{3}\pi r^2 h~~$ for $r$
2. $d=h-\frac{1}{2}gt^2~~$ for $t$

Activity 3.1.3. Compound Interest.

Cyril would like to invest $5000 for two years in a money market account that pays interest compounded annually.

Write a formula for the balance, $B\text{,}$ of Cyril’s account after two years in terms of the interest rate, $r\text{.}$
Complete the table showing Cyril’s account balance after two years for various interest rates.

$r$ $0.02$ $0.04$ $0.06$ $0.08$

$B$ $\hphantom{000000}$ $\hphantom{000000}$ $\hphantom{000000}$ $\hphantom{000000}$
If Cyril would like to have $6250 in two years, what interest rate must the account pay?

Watch out for the mistake of solving $A=P(1+r)^2$ by subtracting $P$ from both sides.
Use technology to graph the formula for Cyril’s account balance. (Use the table of values to help you choose a window.) Sketch your graph on paper, including scales on the axes). Label the coordinates of the point on the graph that corresponds to the filter in part (c).

Subsection 3.1.1 Check Your Understanding

Sketch a graph that illustrates why a quadratic equation can have two solutions.
In part (d) of Activity 1, what technique did you use to solve the equation?
Explain how to deal with the fractions in part (3) of Activity 2.
Explain how you found the solution to part (c) of Activity 3, being careful to list all the steps you used.

Subsection 3.1.2 Wrap Up

In this Lesson, we worked on the following skills and goals related to quadratic models:

Recognize a quadratic equation and its parameters
Graph a parabola by plotting points
Solve a quadratic equation by extraction of roots
Use formulas that involve quadratic terms

Subsection 3.1.3 Questions for Writing or Discussion

Use graphs to explain why a linear equation can have only one solution, but a quadratic equation may have two solutions.
Explain the difference between the volume and the surface area of a cylinder. Include units in your explanation.
Which of these is an exact value: $\sqrt{5}$ or $2.236067977\text{?}$
Delbert squared $-6$ by entering $-6^2$ into his calculator, and got $-36\text{.}$ What went wrong?

Concept Questions.

Which of the following equations cannot be solved by extraction of roots?
1. $\displaystyle 2x^2+3=8$
2. $\displaystyle 2x^2+3x=8$
3. $\displaystyle 2(x+3)^2=8$
4. $\displaystyle 2x^2-3=0$
Which solutions are approximations?
1. $\displaystyle x^2=40,~~~~x= \pm 6.32455532$
2. $\displaystyle t^2 = \dfrac{81}{64},~~~~t= \pm 1.125$
3. $\displaystyle w^2=50,~~~~w= \pm 5\sqrt{2}$
4. $\displaystyle b^2 = (0.632)^2,~~~~b= \pm 0.632$
What is the first step in solving the equation $~2(x+1)^2 = 17\text{?}$
1. Expand $(x+1)^2\text{.}$
2. Get zero on one side.
3. Divide both sides by 2.
4. Take the square root of both sides.
Which statement is true?
1. A quadratic equation may include a linear or a constant term.
2. The solutions of a quadratic equation are always of the form $\pm k\text{.}$
3. Your calculator gives exact decimal values for square roots of integers.
4. The coefficients of a quadratic equation are called parabolas.

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\(r\) (cm)	1	2	3	4	5	6	7	8
\(V\) (cm\(^3\))	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)

\(r\)	\(0.02\)	\(0.04\)	\(0.06\)	\(0.08\)
\(B\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)	\(\hphantom{000000}\)