Chapter 1 Flashcards
Zero-factor property
If a and b are complex numbers with ab = 0, then a = 0 or b = 0 or both equal zero.
Square root property
If x^2 = k, then x = √k or x = -√k
Solving a Quadratic Equation by Completing the Square
To solve ax^2 + bx + x = 0, where a does not equal 0, by completing the square, use these steps.
- If a does not equal 1, divide both sides of the equation by a.
- Rewrite the equation so that the constant term is alone on one side of the equality symbol.
- Square half the coefficient of x, and add this square to each side of the equation.
- Factor the resulting trinomial as a perfect square and combine like terms on the other side.
- Use the square root property to complete the solution,
Quadratic formula
The solutions of the quadratic equation ax^2 + bx + c = 0, where a does not equal 0, are given by the quadratic formula.
x = -b +- √b^2 - 4ac/2a
The Discriminant
Discriminant: Positive, perfect square
Number of Solutions: two
Types of Solutions: rational
Discriminant: positive, but not a perfect square
Number of Solutions: two
Types of Solutions: irrational
Discriminant: zero
Number of Solutions: one (a double solution)
Types of Solutions: rational
Discriminant: negative
Number of Solutions: two (imaginary)
Types of Solutions: nonreal complex (imaginary)
Steps for Problem Solving
A. Read the problem B. Make a sketch or drawing C. Assign a variable name. D. Write an equation. E. Solve the equation. F. State the answer. G. Check
Rational Equations
A rational equation is an equation that has a rational expression for one or more terms. Because a rational expression is not defined when its denominator is 0, values of the variable for which any denominator equals 0 cannot be solutions of the equation. To solve a rational equation, begin by multiplying both sides by the least common denominator of the terms of the equation.
Work Rate Problems
If a job can be done in t units of time, then the rate of work is 1/t of the job per time unit. Therefore, rate * time = portion of the job completed
If the letters r, t, and A represent the rate at which work is done, the time, and the amount of work accomplished, respectively, then A = r t.
Amounts of work are often measured in terms of the number of jobs accomplished. For instance, if one job is accomplished in t time units, then A = 1 and
r = 1/t
Linear Inequality in One Variable
A linear inequality in one variable is an inequality that can be written in the form ax + b > 0, where a and b are real numbers, with a ≠ 0. (Any of the symbols ≥, >,
Solving a Quadratic Inequality
- Factor if necessary
- Apply the zero-factor property and solve for x-values.
- Choose test points based on your x-values.
- State solution set.
Quadratic Inequalities
A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c = 0 for real numbers a, b, and c, with a ≠ 0. (The real symbol , ≤, or ≥.)
Solving a Rational Inequality
- Rewrite the inequality, if necessary, so that 0 is on one side and there is a single fraction on the other side.
- Determine the values that will cause either the numerator or the denominator of the rational expression to equal 0. These values determine the intervals of the number line to consider.
- Use a test value from each interval to determine which intervals form the solution set.
A value causing the denominator to equal zero will never be included in the solution set. If the inequality is strict, any value causing the numerator to equal will be excluded. If the inequality is nonstrict, any such value will be included.
Absolute Value
A number’s distance from 0 on the number line. Example: |-10| means the absolute value of -10 since -10 is 10 units away from 0 on the number line
Three Cases for Absolute Value Equations and Inequalities
Absolute Value Equation or Inequality: Case 1: |x| = k
Equivalent Form: x = k or x = -k
Solution Set: {-k, k}
Absolute Value Equation or Inequality: Case 2: |x| k
Equivalent Form: x k
Solution Set: {-x, -k} ∪ {k, x}
