Overview
This lecture explains quadratic equations, focusing on standard form, identifying coefficients, and rearranging various forms into the standard quadratic equation structure.
Definition and Standard Form of Quadratic Equations
- A quadratic equation in one variable is a second-degree equation (highest exponent is 2).
- The standard form is ( ax^2 + bx + c = 0 ), where ( a, b, ) and ( c ) are real numbers and ( a \neq 0 ).
- If ( a = 0 ), the equation becomes linear, not quadratic.
Parts of a Quadratic Equation
- ( ax^2 ) is called the quadratic term.
- ( bx ) is the linear term.
- ( c ) is the constant term.
Identifying Coefficients ( a, b, c )
- Example: ( x^2 - 5x + 3 = 0 ) ⇒ ( a=1, b=-5, c=3 ).
- ( 4m^2 + 4m + 1 = 0 ) ⇒ ( a=4, b=4, c=1 ).
- ( 9r^2 - 25 = 0 ) ⇒ ( a=9, b=0, c=-25 ).
- ( \frac{1}{2}x^2 + 3x = 0 ) ⇒ ( a=\frac{1}{2}, b=3, c=0 ).
Writing Equations in Standard Form
- Rearranging: Move all terms to one side so the equation equals zero.
- Change the sign when moving terms across the equal symbol.
More Standard Form Examples
- ( x^2 + x = 4 ) ⇒ ( x^2 + x - 4 = 0 ) (( a=1, b=1, c=-4 )).
- ( 7x^2 = \frac{1}{3}x ) ⇒ ( 7x^2 - \frac{1}{3}x = 0 ) (( a=7, b=-\frac{1}{3}, c=0 )).
- ( 6x^2 = 9 ) ⇒ ( 6x^2 - 9 = 0 ) (( a=6, b=0, c=-9 )).
- ( -8x^2 + x = 6 ) ⇒ Multiply by (-1): ( 8x^2 - x - 6 = 0 ) (( a=8, b=-1, c=6 )).
Expanding and Standardizing Product Form Equations
- Expand and combine like terms before arranging to standard form.
- ( 3x(x-2) = 10 ) ⇒ ( 3x^2 - 6x - 10 = 0 ) (( a=3, b=-6, c=-10 )).
- ( (2x+5)(x-1) = 6 ) ⇒ ( 2x^2 + 3x - 5 = 6 ); move 6, combine constants: ( 2x^2 + 3x + 1 = 0 ) (( a=2, b=3, c=1 )).
Properties and Checks on Quadratic Equations
- If ( a = 0 ), the equation is not quadratic (becomes linear).
- If ( b = 0 ) or ( c = 0 ), the equation can still be quadratic as long as ( a \neq 0 ).
Key Terms & Definitions
- Quadratic Equation — An equation of the form ( ax^2 + bx + c = 0 ) with ( a \neq 0 ).
- Quadratic Term — The ( ax^2 ) part of the equation.
- Linear Term — The ( bx ) part of the equation.
- Constant Term — The ( c ) part of the equation.
- Standard Form — The arrangement ( ax^2 + bx + c = 0 ).
Action Items / Next Steps
- Practice rearranging given equations into standard form.
- Identify ( a, b, ) and ( c ) from various quadratic equations.
- Review examples and try similar problems for mastery.