Describing Quadratic Equations
A quadratic equation is an equation that has a second-degree term and no higher terms. A second-degree term is a variable raised to the second power, like x2. When you graph a quadratic equation, you get a parabola, and the solutions to the quadratic equation represent where the parabola crosses the x-axis.
A quadratic equation can be written in the form:
quadratic equation, where a, b, and c are numbers (a ≠0), and x is the variable. x is a solution (or a root) if it satisfies the equation ax2 + bx + c = 0.
Some examples of quadratic equations include:
3x2 + 9x - 2 = 0 6x2 + 11x = 7 4x2 = 13
--------------------------------------------------------------------------------
Solving a Quadratic Formula: Some quadratic equations can be solved easily by factoring. Some simple-to-solve quadratic equations are:
x2 - 1 = 0 (x + 1)(x - 1) = 0
x = ±1 x2 - 5x + 6 = 0 (x - 2)(x - 3) = 0
x = 2, 3
Most second-degree equations are more difficult to solve, and cannot be solved by simple factoring. The quadratic formula is a general way of solving any quadratic equation:
quadratic formula
This formula gives two solutions, although the two solutions may be the same number. (When solving any polynomial equation of degree n, there are at most n solutions to that equation.)
--------------------------------------------------------------------------------
Deriving the Quadratic Formula: The quadratic formula is obtained by solving the general quadratic equation. This is one way to derive the quadratic formula:
Deriving Quadratic
Divide each side of the equation by a.
Deriving Quadratic Subtract c/a from each side of the equation.
Deriving Quadratic Add (b/2a)2 to each side of the equation (to complete the square).
Deriving Quadratic Find a common denominator for the right side of the equation.
Deriving Quadratic Take the square root of each side of the equation.
Deriving Quadratic Add b/2a to each side
A quadratic equation can be written in the form:
quadratic equation, where a, b, and c are numbers (a ≠0), and x is the variable. x is a solution (or a root) if it satisfies the equation ax2 + bx + c = 0.
Some examples of quadratic equations include:
3x2 + 9x - 2 = 0 6x2 + 11x = 7 4x2 = 13
--------------------------------------------------------------------------------
Solving a Quadratic Formula: Some quadratic equations can be solved easily by factoring. Some simple-to-solve quadratic equations are:
x2 - 1 = 0 (x + 1)(x - 1) = 0
x = ±1 x2 - 5x + 6 = 0 (x - 2)(x - 3) = 0
x = 2, 3
Most second-degree equations are more difficult to solve, and cannot be solved by simple factoring. The quadratic formula is a general way of solving any quadratic equation:
quadratic formula
This formula gives two solutions, although the two solutions may be the same number. (When solving any polynomial equation of degree n, there are at most n solutions to that equation.)
--------------------------------------------------------------------------------
Deriving the Quadratic Formula: The quadratic formula is obtained by solving the general quadratic equation. This is one way to derive the quadratic formula:
Deriving Quadratic
Divide each side of the equation by a.
Deriving Quadratic Subtract c/a from each side of the equation.
Deriving Quadratic Add (b/2a)2 to each side of the equation (to complete the square).
Deriving Quadratic Find a common denominator for the right side of the equation.
Deriving Quadratic Take the square root of each side of the equation.
Deriving Quadratic Add b/2a to each side