Questions and Answers
When raising a negative number to a power, what should you always do?
What is the procedure for factoring ax² + bx + c?
Factoring ax² + bx + c involves breaking down the expression into simpler binomials.
What is factoring by grouping?
Factoring by grouping involves rearranging and grouping terms to simplify the factoring process.
What happens when canceling out opposites?
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How do you factor out a -1?
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What are operations with fractions?
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What are complex fractions?
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What is the first step to solve equations with fractions?
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What is the quadratic formula?
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What does it mean to complete the square?
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What does the discriminant indicate about the roots?
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What are the formulas for the sum and product of the roots?
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How do you solve absolute value inequalities?
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Study Notes
Raising a Negative Number to a Power
- Always enclose the negative number in parentheses to avoid confusion with exponents.
Factoring ax²+bx+c
- Use methods such as trial and error, grouping, or applying the quadratic formula to find factors.
Factoring by Grouping
- Group terms to create pairs; factor out the common factors to simplify the expression.
Canceling Out Opposites
- Opposite sign terms can be crossed out, replacing them with -1 in the simplified equation.
Factoring Out a -1
- Remove the negative sign and change the signs of all terms in the expression to reflect the factorization.
Operations with Fractions
- Perform addition, subtraction, multiplication, and division while finding common denominators as needed.
Complex Fractions
- Simplify complex fractions by multiplying the numerator and denominator by the least common denominator to eliminate the fractions.
Equations with Fractions
- Find a common denominator, eliminate the denominators, solve the resulting equation, and check for extraneous roots.
Quadratic Formula (Finding the Roots)
- The formula is expressed as x = (-b ± √(b² - 4ac)) / (2a) to calculate the roots of a quadratic equation.
Completing the Square
- Convert a quadratic equation into the form (x - p)² = q to easily identify the roots.
Discriminant (Describing the Roots)
- A discriminant of 0 indicates equal roots, negative indicates complex roots, a perfect square indicates real and rational roots, and a non-perfect square indicates real and irrational roots.
Sum and Product of Roots
- The sum of the roots is -b/a and the product is c/a. The quadratic equation can be expressed as x² - (sum)x + (product) = 0.
Absolute Value Inequalities
- Isolate the absolute value, reverse the inequality sign and change the right side signs. Utilize “or” between solutions for greater than/greater than or equal to, and “and” for less than/less than or equal to scenarios.
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