College Algebra Test 1 Flashcards
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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 (B)

Is the equation 3/x - 9 = 0 linear in one variable?

False (B)

Is the equation 2x + 4y = 5 linear in one variable?

<p>False (B)</p> Signup and view all the answers

Is the equation x/3 - 9 = 0 linear in one variable?

<p>False (B)</p> Signup and view all the answers

What is the solution for y in the equation (y + 5)/2 - (y - 2)/4 = (y + 7)/3 + 1?

<p>y = -4</p> Signup and view all the answers

What is the value of m in the equation 5/(m - 2) = 3m/(m^2 + 2m - 8) - 2/(m + 4)?

<p>m = -4</p> Signup and view all the answers

What is the expression for b when solving the equation A = 1/2h(B + b)?

<p>b = (2A)/h - B or (2A - Bh)/h</p> Signup and view all the answers

Kent invested a total of $8000. How much did he invest in the stock fund and the CD?

<p>Stock fund = $6000, CD = $2000</p> Signup and view all the answers

How much 4% acid solution should be mixed with 200 mL of a 12% acid solution to make a 9% acid solution?

<p>120 mL 4% acid solution</p> Signup and view all the answers

What is the square root of -12?

<p>2i sqrt(3)</p> Signup and view all the answers

What is the value of the square root of -15 times the square root of -3?

<p>-3 sqrt(5)</p> Signup and view all the answers

What is the result of square root of -50 divided by square root of -2?

<p>5</p> Signup and view all the answers

What is the real part of the complex number 3 + 4i?

<p>3</p> Signup and view all the answers

What is the imaginary part of the complex number (1/2) - (2/3)i?

<p>-2/3</p> Signup and view all the answers

What is the standard form of the complex number 3 - sqrt(-100)?

<p>3 - 10i</p> Signup and view all the answers

What is the simplified form of (-6 + sqrt(-18))/9?

<p>-(2/3) + (sqrt(2))/3 i</p> Signup and view all the answers

What is the value of i raised to the power of -19?

<p>i</p> Signup and view all the answers

What is the value of i raised to the power of 23?

<p>-i</p> Signup and view all the answers

How much 10% solution needs to be added to 40cc of 50% solution to make a 35% solution?

<p>24cc</p> Signup and view all the answers

A total amount of $7000 was added into a stock fund and a CD. How much was added into each account?

<p>Stock fund = $500, CD = $6500</p> Signup and view all the answers

What is the sum of the expression (-2 - 4i) + (5 + 2i) - (3 - 6i)?

<p>4i</p> Signup and view all the answers

What is the result of (a + bi)(a - bi)?

<p>a^2 + b^2</p> Signup and view all the answers

What is the result of (8 + 2i)/(3 + 5i)?

<p>1 - i</p> Signup and view all the answers

What are the roots of the equation n^2 + 5n = 24?

<p>{-8, 3}</p> Signup and view all the answers

What are the solutions for x in the equation x^2 = 64?

<p>{±8}</p> Signup and view all the answers

What is the solution for w in the equation (w + 3)^2 = 8?

<p>{-3 ± 2 sqrt(2)}</p> Signup and view all the answers

In the formula S = n/2(a + d), what is the expression to solve for d?

<p>d = 2S/n - a</p> Signup and view all the answers

How much E5 should be mixed with 5000 gallons of E10 to make an E9 mixture?

<p>1250 gallons</p> Signup and view all the answers

What is the result of (6 + 2i)/(3 - i)?

<p>8/5 + 6/5 i</p> Signup and view all the answers

For the quadratic expression x^2 + 2/9x + n, what value of n makes it a perfect square and what are its factors?

<p>n = 1/81, factors = (x + 1/9)^2</p> Signup and view all the answers

What is the solution for z in the equation (3z - 18)^2 + 58 = 40?

<p>6 + i sqrt(2)</p> Signup and view all the answers

What is the process to make a polynomial a perfect square?

<p>Divide (b) by 2, and square the result.</p> Signup and view all the answers

What is the quadratic formula?

<p>X = -b ± √(b² - 4ac) / (2a)</p> Signup and view all the answers

Use the quadratic formula to solve the equation 5x^2 - 6x + 3.

<p>3/5 ± √(6)/5 i</p> Signup and view all the answers

What is the discriminant in quadratic equations?

<p>b² - 4ac</p> Signup and view all the answers

If the discriminant is negative, how many solutions are there?

<p>False (B)</p> Signup and view all the answers

If the discriminant is positive, how many solutions are there?

<p>True (A)</p> Signup and view all the answers

If the discriminant is zero, how many solutions are there?

<p>True (A)</p> Signup and view all the answers

What is the absolute value of -7?

<p>{}</p> Signup and view all the answers

What are the solutions for the absolute value of 3?

<p>{-3, 3}</p> Signup and view all the answers

What are the solutions for the equation 5|2 - 4t| = 50?

<p>{-2, 3}</p> Signup and view all the answers

What are the solutions for the equation |3x - 4| = |2x + 1|?

<p>{5, 3/5}</p> Signup and view all the answers

If 0 = 0, it means it is an identity and pertains to all real numbers.

<p>True (A)</p> Signup and view all the answers

What is the solution for x in the equation 2(x - 4)^(3/4)?

<p>x = 85</p> Signup and view all the answers

What is the inequality for m in the equation (m - 4)/2 - (3m + 4)/10 > -3/5m?

<p>m &gt; 3, set builder notation: {m | m &gt; 3}, interval notation: (3, ∞)</p> Signup and view all the answers

Write the set builder notation and interval notation for y < 2.

<p>{y | y &lt; 2}, (-∞, -4) U (2, ∞)</p> Signup and view all the answers

Flashcards

Linear equations in one variable

Have the form ax + b = 0, where a and b are constants.

Non-linear equation example

An equation that does not form a straight line, such as 6x² + 1 = 0.

Fractions in linear equations

Fractions can make equations non-linear, as in 3/x - 9 = 0.

Multi-variable equation

An equation with two or more variables, e.g., 2x + 4y = 5.

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Linear equation example

An example of a linear equation in one variable is x/3 - 9 = 0.

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Solving linear equations

Finding the value of the variable that makes the equation true.

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Area of a trapezoid formula

The formula is A = 1/2 h (B + b), where h is height and B, b are bases.

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Investment interest calculation

Total interest from investments is calculated based on principal and rate.

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Creating acid solutions

Mix different concentrations to achieve desired acid strength, e.g., 200 mL of 12% plus 120 mL of 4% gives 9%.

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Complex numbers definition

Numbers that include the imaginary unit i, where i = √-1.

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Square root of negative

The square root of -12 is expressed as 2i√3.

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Quadratic equation solutions

Solutions to equations like n² + 5n - 24 = 0 found using factoring or formulas.

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Quadratic formula

Formula used to solve ax² + bx + c = 0: X = (-b ± √(b² - 4ac))/(2a).

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Discriminant in quadratics

The discriminant (b² - 4ac) determines the number and type of solutions.

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Absolute value definition

The absolute value of a number is its distance from zero, e.g., |−7| = 7.

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Solving absolute value equations

Equations like 5|2 - 4t| = 50 yield two potential solutions.

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Identity equation

An equation that is true for all values, like 0 = 0.

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Set-builder notation

A notation that describes a set by stating the properties of its elements, e.g., {m | m > 3}.

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Interval notation

A way to denote sets of numbers using intervals, e.g., (3, ∞) means all numbers greater than 3.

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Simplifying inequalities

Finding limits for variables, such as simplifying m - 4 > 3/5m leads to m > 3.

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Real and imaginary parts

In complex numbers, the real part is the number without i, while the imaginary part includes it.

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Completing the square

A method to solve quadratic equations by making a perfect square from the expression.

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Nature of solutions

Determined by the discriminant: positive (two real), zero (one real), negative (no real solutions).

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Absolute value solutions

|3x - 4| = |2x + 1| results in two distinct solutions.

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Mixed solutions in mathematics

Different concentrations or types of solutions can be combined to achieve new outcomes.

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Multi-variable equations

Equations that involve multiple variables indicating the relationship between them, e.g., 2x + 4y = 5.

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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) ).

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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.

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