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Mathematics LibreTexts

3.1.1: Exercises 3.1

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    63461
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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


    3.1.1: Exercises 3.1 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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