Overview
This lecture introduces quadratic equations, compares them to linear equations, demonstrates their standard forms, and teaches how to identify and write quadratic equations from word problems.
Linear vs. Quadratic Equations
- Linear equations have the highest variable exponent of 1 (e.g., 9x + 15 = 91).
- Quadratic equations have the highest variable exponent of 2 (e.g., 5x² - 23x = 19).
Standard Form of Quadratic Equations
- The standard form is ax² + bx + c = 0, where a ≠0, and a, b, c are real numbers.
- The quadratic term is ax², the linear term is bx, and the constant is c.
- If a = 0, the equation becomes linear.
- The standard form should have a > 0 and coefficients relatively prime.
Writing Equivalent Standard Forms
- Equations can be rearranged using equality and symmetric properties but should follow ax² + bx + c = 0 with a > 0.
- If a, b, and c share a common factor, divide both sides to simplify (e.g., 3x² + 9x - 12 = 0 becomes x² + 3x - 4 = 0).
Identifying Coefficients
- For each quadratic equation in standard form, identify values of a, b, and c.
- Example: 2x² - 2x + 7 = 0; a = 2, b = -2, c = 7.
Translating Word Problems into Equations
- Convert verbal situations into algebraic equations to determine if they are quadratic.
- Example: "The square of a number plus 8 times the number is 100" becomes x² + 8x = 100, a quadratic equation.
Word Problem Examples
- Credit card balance problem leads to a linear equation (not quadratic).
- Cube with reduced dimensions and given surface area leads to a quadratic equation.
- Area and perimeter of a rectangle problem leads to a quadratic equation.
Key Terms & Definitions
- Linear Equation — An equation with the highest variable exponent of 1.
- Quadratic Equation — An equation with the highest variable exponent of 2.
- Standard Form — The form ax² + bx + c = 0, where a ≠0.
- Coefficient — The numerical factor of a term (a, b, c in standard form).
- Relatively Prime — Coefficients having no common factor other than 1.
Action Items / Next Steps
- Practice writing equations from real-life scenarios and identifying if they are quadratic.
- Review and memorize standard forms and coefficient identification.
- Complete any assigned word problems converting situations to quadratic equations.