Table of contents
- 0. Review of Algebra4h 16m
- 1. Equations & Inequalities3h 18m
- 2. Graphs of Equations43m
- 3. Functions2h 17m
- 4. Polynomial Functions1h 44m
- 5. Rational Functions1h 23m
- 6. Exponential & Logarithmic Functions2h 28m
- 7. Systems of Equations & Matrices4h 6m
- 8. Conic Sections2h 23m
- 9. Sequences, Series, & Induction1h 19m
- 10. Combinatorics & Probability1h 45m
3. Functions
Intro to Functions & Their Graphs
2:44 minutes
Problem 105a
Textbook Question
Textbook QuestionExercises 103–105 will help you prepare for the material covered in the next section. Solve by completing the square: y² – 6y — 4 = 0.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Completing the Square
Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves rearranging the equation and adding a specific value to both sides to create a square of a binomial. This technique simplifies the process of finding the roots of the equation and is particularly useful when the quadratic is not easily factorable.
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Quadratic Equations
A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. Understanding the standard form and properties of quadratic equations is essential for solving them effectively.
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Discriminant
The discriminant is a component of the quadratic formula, given by the expression b² - 4ac. It determines the nature of the roots of a quadratic equation: if the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root (a repeated root); and if it is negative, there are two complex roots. Analyzing the discriminant helps in understanding the solutions without necessarily solving the equation.
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