3.1.1: Exercises 3.1

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Terms and Concepts

Exercise \(\PageIndex{1}\)

Is it possible to solve a cubic statement?

Answer: Yes, but it can be quite difficult, especially if it has many parameters.

Exercise \(\PageIndex{2}\)

What are the possible types of solutions when solving a quadratic statement?

Answer: 2 distinct real roots; 1 repeated real root; a complex conjugate pair of roots.

Exercise \(\PageIndex{3}\)

What is the maximum number of different solutions that a seventh degree statement could have?

Answer: 7

Exercise \(\PageIndex{4}\)

T/F: A cubic statement can have only complex solutions. Explain.

Answer: F; it must have at least one real solution since complex solutions come in pairs

Problems

In exercises \(\PageIndex{5}\) - \(\PageIndex{10}\), determine the type of statement in terms of the given variable.

Exercise \(\PageIndex{5}\)

\(x^3y+2x^2yz-6xz^2 = yz^2 -10\) in terms of \(x\)

Answer: cubic

Exercise \(\PageIndex{6}\)

\(x^3y+2x^2yz-6xz^2 = yz^2 -10\) in terms of \(y\)

Answer: linear

Exercise \(\PageIndex{7}\)

\(x^3y+2x^2yz-6xz^2 = yz^2 -10\) in terms of \(z\)

Answer: quadratic

Exercise \(\PageIndex{8}\)

\(xt + \cos{(\theta)}=x^4t^3-6t\) in terms of \(\theta\)

Answer: trigonometric

Exercise \(\PageIndex{9}\)

\(xt + \cos{(\theta)}=x^4t^3-6t\) in terms of \(x\)

Answer: quartic, or a statement of degree 4

Exercise \(\PageIndex{10}\)

\(xt + \cos{(\theta)}=x^4t^3-6t\) in terms of \(t\)

Answer: cubic

In exercises \(\PageIndex{11}\) - \(\PageIndex{19}\), determine if it is possible to solve the statement for the given variable. If it is possible, solve but do not simplify your answer(s). If it is not possible, explain why.

Exercise \(\PageIndex{11}\)

\(xy^2-xy=5y-3x\) for \(x\)

Answer: It is possible to solve; \(x=\displaystyle \frac{5y}{y^2-y+3}\)

Exercise \(\PageIndex{12}\)

\(xy^2-xy=5y-3x\) for \(y\)

Answer: \(\displaystyle \frac{x+5 \pm\sqrt{-11x^2+10x+25}}{2x}\)

Exercise \(\PageIndex{13}\)

\(3t^2-5mq=8qt+2m^3\) for \(q\)

Answer: It is possible to solve; \(q=\displaystyle \frac{3t^2-2m^3}{8t+5m}\)

Exercise \(\PageIndex{14}\)

\(2a^2bc^3+3abc^2+4a^2c^2-3b=4c\) for \(a\)

Answer: It is possible to solve; \(a = \displaystyle \frac{-(3bc^2) \pm \sqrt{(3bc^2)^2 - 4 (2bc^3+4c^2)(-3b-4c)}}{2(2bc^3+4c^2)}\)

Exercise \(\PageIndex{15}\)

\(2a^2bc^3+3abc^2+4a^2c^2-3b=4c\) for \(b\)

Answer: It is possible to solve; \(b = \displaystyle \frac{4c-4a^2c^2}{2a^2c^3+3ac^2-3}\)

Exercise \(\PageIndex{16}\)

\(2a^2bc^3+3abc^2+4a^2c^2-3b=4c\) for \(c\)

Answer: It is possible to solve; but it would require using the cubic root formula

Exercise \(\PageIndex{17}\)

\(\log_2{(xy)=x+e^z}\) for x

Answer: Not possible to solve for x; it is inside of a logarithm and has a linear term

Exercise \(\PageIndex{18}\)

\(\log_2{(xy)=x+e^z}\) for y

Answer: It is possible to solve; \(\displaystyle y= 2^{x+e^z-log_2{(x)}}\) or \(\displaystyle y= \frac{2^{x+e^z}}{x}\)

Exercise \(\PageIndex{19}\)

\(\log_2{(xy)=x+e^z}\) for z

Answer: It is possible to solve; \(\displaystyle z= \ln{[log_2{(xy)} -x]}\)

In exercises \(\PageIndex{20}\) - \(\PageIndex{28}\), solve for \(x\). Be sure to list all possible values of \(x\).

Exercise \(\PageIndex{20}\)

\(x^2-16=0\)

Answer: \(x=-4,4\)

Exercise \(\PageIndex{21}\)

\(x^2+16=0\)

Answer: \(x=-4i,4i\)

Exercise \(\PageIndex{22}\)

\(x^2-4x-7=2\)

Answer: \(x=2+\sqrt{13}, 2- \sqrt{13}\)

Exercise \(\PageIndex{23}\)

\(x^2-2x+7=2\)

Answer: \(x=1+2i, 1-2i\)

Exercise \(\PageIndex{24}\)

\(5x^2+2x=-1\)

Answer: \(\displaystyle x=\frac{-1+2i}{5}, \frac{-1-2i}{5}\)

Exercise \(\PageIndex{25}\)

\(x^3=8\)

Answer: \(x=2\)

Exercise \(\PageIndex{26}\)

\(x^3+x^2=4x+4\)

Answer: \(x=-2, -1, 2\)

Exercise \(\PageIndex{27}\)

\(2(x-3)^2-7 = -4x+9\)

Answer: \(x=2-\sqrt{3}, 2+ \sqrt{3}\)

Exercise \(\PageIndex{28}\)

\((x+2)^3 = 2x^2+8x+7\)

Answer: \(\displaystyle x=-1, \frac{-3+\sqrt{5}}{2}, \frac{-3-\sqrt{5}}{2}\)

In exercises \(\PageIndex{29}\) - \(\PageIndex{33}\), classify the type(s) of solution(s) from the given exercise.

Exercise \(\PageIndex{29}\)

Exercise 3.1.1.20

Answer: Two real solutions

Exercise \(\PageIndex{30}\)

Exercise 3.1.1.21

Answer: A complex conjugate pair

Exercise \(\PageIndex{31}\)

Exercise 3.1.1.22

Answer: Two real solutions

Exercise \(\PageIndex{32}\)

Exercise 3.1.1.25

Answer: One repeated solution

Exercise \(\PageIndex{33}\)

Exercise 3.1.1.26

Answer: Three real solutions