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3.1.1: Exercises 3.1 - Mathematics


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{( heta)}=x^4t^3-6t) in terms of ( heta)

Answer

trigonometric

Exercise (PageIndex{9})

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

Answer

quartic, or a statement of degree 4

Exercise (PageIndex{10})

(xt + cos{( heta)}=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 pmsqrt{-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


Watch the video: 10th Maths. ExerciseChapter 3. Algebra இயறகணதம Solving Simultaneous linear equations (October 2021).