Questions and Answers
What is the standard form of a linear equation in one variable?
ax + b = 0
Is the equation 6x^2 + 1 = 0 linear in one variable?
False
Is the equation 3/x - 9 = 0 linear in one variable?
False
Is the equation 2x + 4y = 5 linear in one variable?
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Is the equation x/3 - 9 = 0 linear in one variable?
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What is the solution for y in the equation (y + 5)/2 - (y - 2)/4 = (y + 7)/3 + 1?
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What is the value of m in the equation 5/(m - 2) = 3m/(m^2 + 2m - 8) - 2/(m + 4)?
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What is the expression for b when solving the equation A = 1/2h(B + b)?
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Kent invested a total of $8000. How much did he invest in the stock fund and the CD?
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How much 4% acid solution should be mixed with 200 mL of a 12% acid solution to make a 9% acid solution?
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What is the square root of -12?
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What is the value of the square root of -15 times the square root of -3?
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What is the result of square root of -50 divided by square root of -2?
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What is the real part of the complex number 3 + 4i?
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What is the imaginary part of the complex number (1/2) - (2/3)i?
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What is the standard form of the complex number 3 - sqrt(-100)?
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What is the simplified form of (-6 + sqrt(-18))/9?
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What is the value of i raised to the power of -19?
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What is the value of i raised to the power of 23?
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How much 10% solution needs to be added to 40cc of 50% solution to make a 35% solution?
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A total amount of $7000 was added into a stock fund and a CD. How much was added into each account?
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What is the sum of the expression (-2 - 4i) + (5 + 2i) - (3 - 6i)?
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What is the result of (a + bi)(a - bi)?
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What is the result of (8 + 2i)/(3 + 5i)?
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What are the roots of the equation n^2 + 5n = 24?
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What are the solutions for x in the equation x^2 = 64?
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What is the solution for w in the equation (w + 3)^2 = 8?
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In the formula S = n/2(a + d), what is the expression to solve for d?
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How much E5 should be mixed with 5000 gallons of E10 to make an E9 mixture?
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What is the result of (6 + 2i)/(3 - i)?
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For the quadratic expression x^2 + 2/9x + n, what value of n makes it a perfect square and what are its factors?
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What is the solution for z in the equation (3z - 18)^2 + 58 = 40?
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What is the process to make a polynomial a perfect square?
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What is the quadratic formula?
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Use the quadratic formula to solve the equation 5x^2 - 6x + 3.
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What is the discriminant in quadratic equations?
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If the discriminant is negative, how many solutions are there?
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If the discriminant is positive, how many solutions are there?
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If the discriminant is zero, how many solutions are there?
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What is the absolute value of -7?
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What are the solutions for the absolute value of 3?
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What are the solutions for the equation 5|2 - 4t| = 50?
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What are the solutions for the equation |3x - 4| = |2x + 1|?
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If 0 = 0, it means it is an identity and pertains to all real numbers.
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What is the solution for x in the equation 2(x - 4)^(3/4)?
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What is the inequality for m in the equation (m - 4)/2 - (3m + 4)/10 > -3/5m?
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Write the set builder notation and interval notation for y < 2.
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Study Notes
Linear Equations
- Linear equations in one variable have the form ( ax + b = 0 ).
- An example of a non-linear equation: ( 6x^2 + 1 = 0 ).
- Fractions in linear equations can make them non-linear, e.g., ( \frac{3}{x} - 9 = 0 ).
- Multi-variable equations, such as ( 2x + 4y = 5 ), are not linear in one variable.
- Equation ( \frac{x}{3} - 9 = 0 ) is linear in one variable.
Solving Equations
- For ( \frac{y+5}{2} - \frac{y-2}{4} = \frac{y+7}{3} + 1 ), the solution is ( y = -4 ).
- The equation ( \frac{5}{m-2} = \frac{3m}{m^2 + 2m - 8} - \frac{2}{m + 4} ) results in ( m = -4 ) with specific restrictions.
Areas and Investments
- The formula for the area of a trapezoid is ( A = \frac{1}{2}h(B + b) ); solving for ( b ) gives ( b = \frac{2A}{h} - B ).
- Investment scenario: Kent invested $6000 in a stock fund at 6.5% for 2 years and $2000 in a CD at 2.5%, yielding total interest of $855.
Acid Solution Mixtures
- To create a 9% acid solution from 200 mL of a 12% solution, mix with 120 mL of a 4% acid solution.
- To produce a 35% solution from 40cc of a 50% solution, add 24cc of a 10% solution.
Complex Numbers
- The square root of (-12) is ( 2i\sqrt{3} ).
- ( \sqrt{-15} \times \sqrt{-3 = -3\sqrt{5} ).
- The value of ( \sqrt{-50} / \sqrt{-2} = 5 ).
- The real part of ( 3 + 4i ) is 3, and the imaginary part of ( \frac{1}{2} - \frac{2}{3}i ) is (-\frac{2}{3}).
- Standard form for ( 3 - \sqrt{-100} ) is ( 3 - 10i ).
Solving Quadratic Equations
- Solutions for ( n^2 + 5n = 24 ) are (-8) and (3).
- Using the square root method, ( x^2 = 64 ) gives ( { \pm 8 } ).
- Completing the square for ( (w+3)^2 = 8 ) yields ( {-3 \pm 2\sqrt{2}} ).
- The quadratic formula is given by ( X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- Discriminant ( b^2 - 4ac ) determines the nature of solutions: negative for non-real solutions, positive for two real solutions, and zero for one real solution.
Absolute Values and Inequalities
- The absolute value of (-7) is (7), and for (3), it is {-3, 3}.
- The equation ( 5|2 - 4t| = 50 ) results in two solutions: {-2, 3}.
- The equation ( |3x - 4| = |2x + 1| ) has solutions ( {5, \frac{3}{5}} ).
Set Notation and Intervals
- An identity equation ( 0 = 0 ) implies it is valid for all real numbers.
- For ( 2(x - 4)^{3/4} ), solving for ( x ) results in ( x = 85 ).
- The inequality ( \frac{m - 4}{2} - \frac{3m + 4}{10} > -\frac{3}{5}m ) simplifies to ( m > 3 ), with set-builder notation as ( {m | m > 3} ) and interval notation as ( (3, \infty) ).
- For ( y_2 ), set-builder notation is ( {y | y < -4} ) and interval notation is ( (-\infty, -4) \cup (2, \infty) ).
Studying That Suits You
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Description
Test your knowledge on linear equations in one variable with this set of flashcards. Perfect for reviewing the key concepts and differentiating between linear and non-linear equations. Challenge yourself and enhance your understanding of algebraic principles.
