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Questions and Answers
What are the complex roots of the equation $2x^3 + 4x^4 + 11x - 17 = 0$ given that it has 1 real root at $x=1$?
What are the complex roots of the equation $2x^3 + 4x^4 + 11x - 17 = 0$ given that it has 1 real root at $x=1$?
-3±5i/2
What kind of solutions does $x^5 + 2x^4 + 2x^3 = 0$ have?
What kind of solutions does $x^5 + 2x^4 + 2x^3 = 0$ have?
- 3 real, all 0 (correct)
- 2 imaginary (correct)
- x=-1±i (correct)
- All of the above
What is the simplified form of $6(4x + 8y) + 2(3x - 6y)$?
What is the simplified form of $6(4x + 8y) + 2(3x - 6y)$?
30x + 36y
What is the simplified form of $10 ext{√}162 - ext{√}405 - ext{√}200$?
What is the simplified form of $10 ext{√}162 - ext{√}405 - ext{√}200$?
What are the solutions for the equation $|x - 2| = 27$?
What are the solutions for the equation $|x - 2| = 27$?
What is the solution for the system of equations $2x + 5y = 12$ and $x - 3y = -2$?
What is the solution for the system of equations $2x + 5y = 12$ and $x - 3y = -2$?
What is the factorization of $4xy - 3y - 40x + 30$?
What is the factorization of $4xy - 3y - 40x + 30$?
What is the factorization of $2x^2 + 3x - 9$?
What is the factorization of $2x^2 + 3x - 9$?
What is the result of adding $(5+i)-(6-12i)$?
What is the result of adding $(5+i)-(6-12i)$?
What is the solution for the equation $√{2x - 2} + 12 = 14$?
What is the solution for the equation $√{2x - 2} + 12 = 14$?
What is the simplified form of $√{36x^{12}y^8}$?
What is the simplified form of $√{36x^{12}y^8}$?
What is the simplified form of $(4x^8y^8/2x^3y^{10})^2$?
What is the simplified form of $(4x^8y^8/2x^3y^{10})^2$?
What is the y-intercept of the function $y = 2 imes 4^x$?
What is the y-intercept of the function $y = 2 imes 4^x$?
What is the simplified form of $8e^8/4e^{13}$?
What is the simplified form of $8e^8/4e^{13}$?
What is the value of $log_2(8)$?
What is the value of $log_2(8)$?
Express $log_r(12) + log_r(45)$ as a single logarithm.
Express $log_r(12) + log_r(45)$ as a single logarithm.
What is the solution for $1/16 = 64^{3x+5}$?
What is the solution for $1/16 = 64^{3x+5}$?
Find x to the nearest 0.01 given $1.95^x = 26$.
Find x to the nearest 0.01 given $1.95^x = 26$.
What is the solution to the equation $e^{0.05t} = 10$?
What is the solution to the equation $e^{0.05t} = 10$?
If the number of bacteria, b, after t minutes is given by the equation $b = 10e^kt$, and there are 593 bacteria after 9 minutes, what is k?
If the number of bacteria, b, after t minutes is given by the equation $b = 10e^kt$, and there are 593 bacteria after 9 minutes, what is k?
What is the simplified form of $(x^2 - 2x - 15)/(x^2 - x - 20)$ if possible?
What is the simplified form of $(x^2 - 2x - 15)/(x^2 - x - 20)$ if possible?
What is the simplified form of $6d^4/3e^3f × 15e^7f^2/20d^9$?
What is the simplified form of $6d^4/3e^3f × 15e^7f^2/20d^9$?
What is the product of $(x - 5)(x + 2)/(x^2 - 25)$?
What is the product of $(x - 5)(x + 2)/(x^2 - 25)$?
What is the simplified result of $x^2 - 4 / x + 2 × 6x^2/2x - 4$?
What is the simplified result of $x^2 - 4 / x + 2 × 6x^2/2x - 4$?
What is the solution for the equation $(x^2 + 9x + 20)/(x^2 - 25) ÷ (x + 4)/(x - 4)$?
What is the solution for the equation $(x^2 + 9x + 20)/(x^2 - 25) ÷ (x + 4)/(x - 4)$?
What is the result of adding $-2x - 6 / 9x + x + 6 / 9x$?
What is the result of adding $-2x - 6 / 9x + x + 6 / 9x$?
What is the result of adding $5/x + 4 + 2/x - 4$?
What is the result of adding $5/x + 4 + 2/x - 4$?
What is the simplified form of $5/x + 5 / 1/x - 6$?
What is the simplified form of $5/x + 5 / 1/x - 6$?
What are the solutions for the equation $2/x - 5 = x - 6$?
What are the solutions for the equation $2/x - 5 = x - 6$?
What is the solution for the equation $x^2/(x + 9) = 81/(x + 9)$?
What is the solution for the equation $x^2/(x + 9) = 81/(x + 9)$?
What is the solution for the equation $x + 1/5x - 2/4x = 1/20$?
What is the solution for the equation $x + 1/5x - 2/4x = 1/20$?
What is the focus of the parabola given by $x = 1/32y^2$?
What is the focus of the parabola given by $x = 1/32y^2$?
What is the equation in vertex form of the parabola with vertex at (0,0) and focus at (5,0)?
What is the equation in vertex form of the parabola with vertex at (0,0) and focus at (5,0)?
What conic section is represented by the equation $x^2 + 4y^2 = 4$?
What conic section is represented by the equation $x^2 + 4y^2 = 4$?
Write the equation of an ellipse, in standard form, with vertices at (-7,0) and (7,0) and co-vertices at (0,-4) and (0,4).
Write the equation of an ellipse, in standard form, with vertices at (-7,0) and (7,0) and co-vertices at (0,-4) and (0,4).
What conic section is represented by the equation $x^2/25 - y^2/9 = 1$?
What conic section is represented by the equation $x^2/25 - y^2/9 = 1$?
Find the coordinates of the center and radius for the circle defined by $x^2 + y^2 - 4x + 4y - 17 = 0$.
Find the coordinates of the center and radius for the circle defined by $x^2 + y^2 - 4x + 4y - 17 = 0$.
What are the solutions for the equation $x^2 - 12x = 28$?
What are the solutions for the equation $x^2 - 12x = 28$?
What is the value of $i^{51}$?
What is the value of $i^{51}$?
Given $f(x) = x^3 + 4x^2 + 2x - 3$, find $f(-20)$.
Given $f(x) = x^3 + 4x^2 + 2x - 3$, find $f(-20)$.
What is the simplified form of $(3x^2 - 12)/(4x^2 + 8x) / (6x - 12)/(8x^2 + 16x)$?
What is the simplified form of $(3x^2 - 12)/(4x^2 + 8x) / (6x - 12)/(8x^2 + 16x)$?
Find the product of $(8 - 2i)(5 + i)$.
Find the product of $(8 - 2i)(5 + i)$.
Solve for the equation $log_2(x) + log_2(x-2) = log_2(3)$.
Solve for the equation $log_2(x) + log_2(x-2) = log_2(3)$.
Solve $4^3x^{-1} = 1/256$.
Solve $4^3x^{-1} = 1/256$.
Simplify the expression $(3x^2/4y)^{-3}$.
Simplify the expression $(3x^2/4y)^{-3}$.
How many positive zeroes does the polynomial $5x^2 - 3x - 2$ have?
How many positive zeroes does the polynomial $5x^2 - 3x - 2$ have?
What are the positive rational zeroes for $5x^2 - 3x - 2$?
What are the positive rational zeroes for $5x^2 - 3x - 2$?
What are the zeroes of the polynomial $5x^2 - 3x - 2$?
What are the zeroes of the polynomial $5x^2 - 3x - 2$?
Write in logarithmic form: $3^3 = 27$.
Write in logarithmic form: $3^3 = 27$.
What is the domain of a polynomial function?
What is the domain of a polynomial function?
Given the restricted domain for the polynomial function $f(x) = 2x^3 - 3x^2 - 2x + 1$ as $-3≤x≤3$, what is the range of the function?
Given the restricted domain for the polynomial function $f(x) = 2x^3 - 3x^2 - 2x + 1$ as $-3≤x≤3$, what is the range of the function?
Which expression is equivalent to $3log(4) + log(6) - log(48)$ when written as a single log expression?
Which expression is equivalent to $3log(4) + log(6) - log(48)$ when written as a single log expression?
Solve the equation $2^x + 1 - 8^x - 2$.
Solve the equation $2^x + 1 - 8^x - 2$.
Solve the equation $ln(2x - 1) = 2$.
Solve the equation $ln(2x - 1) = 2$.
Solve for $x$ in the equation $-2x^3 + 14x^2 - 24x + 20 = 0$.
Solve for $x$ in the equation $-2x^3 + 14x^2 - 24x + 20 = 0$.
Find the equation of the polynomial with roots $5, -4, 1$.
Find the equation of the polynomial with roots $5, -4, 1$.
Find the roots of the polynomial equation $x^3 - 2x^2 + 10x + 136 = 0$.
Find the roots of the polynomial equation $x^3 - 2x^2 + 10x + 136 = 0$.
Rationalize the denominator of $ rac{√3 - √6}{√3 + √6}$.
Rationalize the denominator of $ rac{√3 - √6}{√3 + √6}$.
What is the center of the conic given by $ rac{(x-3)^2}{36} - rac{(y+2)^2}{81} = 1$?
What is the center of the conic given by $ rac{(x-3)^2}{36} - rac{(y+2)^2}{81} = 1$?
Solve for $x$ in the equation $-6x - 4x = 6 + 5x$.
Solve for $x$ in the equation $-6x - 4x = 6 + 5x$.
Solve for $x$ in the equation $-1/2 - 4x/4 = 1/2 - 2x$.
Solve for $x$ in the equation $-1/2 - 4x/4 = 1/2 - 2x$.
Solve for $x$ in the equation $8x^2 = 12x - 4$.
Solve for $x$ in the equation $8x^2 = 12x - 4$.
Solve for $x$ in the equation $-5x^2 - 2 = 14x$.
Solve for $x$ in the equation $-5x^2 - 2 = 14x$.
Solve the following system of equations: $-3x - 9y = 7$, $8x + 9y = -1$.
Solve the following system of equations: $-3x - 9y = 7$, $8x + 9y = -1$.
Solve for $x$ in the equation $-2 - 2x = 2 + 2x/5$.
Solve for $x$ in the equation $-2 - 2x = 2 + 2x/5$.
What is the slope of the line given by the equation $y = rac{3}{5}x - 7$?
What is the slope of the line given by the equation $y = rac{3}{5}x - 7$?
In the arithmetic series described by $a_n = a_1 + (n - 1)d$, what does the variable $n$ represent?
In the arithmetic series described by $a_n = a_1 + (n - 1)d$, what does the variable $n$ represent?
Given the arithmetic series $-3, -3/2, 0, 3/2,...$, what is the common difference?
Given the arithmetic series $-3, -3/2, 0, 3/2,...$, what is the common difference?
What is the value of $log_2(16)$?
What is the value of $log_2(16)$?
Find the sum of the arithmetic sequence with $a_1 = 60$, $d = 4$, $a_n = 120$.
Find the sum of the arithmetic sequence with $a_1 = 60$, $d = 4$, $a_n = 120$.
What is the 80th term of the arithmetic sequence: $-1, 1, 3,...$?
What is the 80th term of the arithmetic sequence: $-1, 1, 3,...$?
Find the sum of the arithmetic sequence: $8, rac{33}{4}, rac{17}{2}, rac{35}{4},..., 50$.
Find the sum of the arithmetic sequence: $8, rac{33}{4}, rac{17}{2}, rac{35}{4},..., 50$.
Given the arithmetic series: $a_n = -3(1/2)$, what is the common ratio?
Given the arithmetic series: $a_n = -3(1/2)$, what is the common ratio?
Find the 5th term for the geometric series given by $a_1 = 2$, $r = 3$.
Find the 5th term for the geometric series given by $a_1 = 2$, $r = 3$.
For the series $∑_{k=1}^∞ 3(2/3)^k$, does this series converge and what is the sum?
For the series $∑_{k=1}^∞ 3(2/3)^k$, does this series converge and what is the sum?
What is the condition for an infinite geometric series to converge?
What is the condition for an infinite geometric series to converge?
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Study Notes
Polynomial Roots and Complex Numbers
- Real and Complex Roots: The equation (2x^3 + 4x^4 + 11x - 17 = 0) has one real root at (x=1) and complex roots given by (-\frac{3}{2} \pm \frac{5i}{2}).
- Types of Solutions: The polynomial (x^5 + 2x^4 + 2x^3 = 0) has three real solutions (including multiplicity) at (x=0) and two imaginary solutions.
Simplification Techniques
- Expression Simplification: Simplifying (6(4x+8y)+2(3x-6y)) yields (30x + 36y).
- Radicals and Square Roots: The expression (10\sqrt{162} - \sqrt{405} - \sqrt{200}) simplifies to (80\sqrt{2} - 9\sqrt{5}).
Solving Equations
- Absolute Value: The equation (|x-2| = 27) leads to two solutions: (x = 29) and (x = -25).
- System of Equations: Solving (2x + 5y = 12) and (x - 3y = -2) gives (x = \frac{26}{11}) and (y = \frac{16}{11}).
Factoring Polynomials
- Quadratic Factoring: The expression (4xy - 3y - 40x + 30) factors to ((4x-3)(y-10)).
- Quadratic Expression: The equation (2x^2 + 3x - 9) factors to ((2x-3)(x+3)).
Operations with Complex Numbers
- Complex Addition: Adding complex numbers ((5+i)-(6-12i)) results in (-1 + 13i).
Rational Expressions
- Simplifying Expressions: (\frac{x^2 - 2x - 15}{x^2 - x - 20}) simplifies to (\frac{x+3}{x+4}).
- Multiplying Rational Expressions: (\frac{x-5}{x^2-25} \cdot \frac{6x^2}{2x-4}) results in (3x^2).
Logarithmic Operations
- Logarithm Evaluation: ( \log_2(8) ) evaluates to (x=3).
- Single Log Expression: Combining logarithms, (\log_r(12) + \log_r(45)) simplifies to (\log_r(540)).
Conic Sections and Parabola Characteristics
- Circle Characteristics: The circle (x^2 + y^2 - 4x + 4y - 17 = 0) has a center at ((2, -2)).
- Parabola Focus: For the parabola (x=\frac{1}{32}y^2), the focus is at ((8, 0)).
Arithmetic and Geometric Sequences
- Arithmetic Series Identification: The series (-3, -\frac{3}{2}, 0, \frac{3}{2},...) has a common difference of (\frac{3}{2}).
- Geometric Series Sum: An infinite geometric series given by (\sum_{k=1}^{\infty} 3\left(\frac{2}{3}\right)^k) converges to (6).
Rationalizing Denominators
- Rationalization Result: The expression (\frac{\sqrt{3}-\sqrt{6}}{\sqrt{3}+\sqrt{6}}) rationalizes to (-3 + 2\sqrt{2}).
Solving Polynomial Equations
- Roots of Polynomial: The equation (-2x^3 + 14x^2 - 24x + 20 = 0) yields roots (5, 1+i, 1-i).
- Polynomial Equation: The polynomial with roots (5, -4, 1) is given by (x^3 - 2x^2 - 19x + 20 = 0).
General Knowledge
- Domain of Polynomial Functions: The domain of a polynomial function includes all real numbers.
- Simple Linear Equations: In linear equations like (-6x - 4x = 6 + 5x), the solution is (x = -\frac{2}{5}).
These notes encapsulate key mathematical concepts and problem-solving techniques relevant to Algebra 2, providing a reference guide for key theorems and solutions.
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