Quadratic Equations C.A.1-3

Solving Quadratic Equations If a quadratic equation ax²+bx+c=0 can’t be solved by factoring that means the solutions involve roots. The quadratic formula can be used to solve any quadratic equation by using the coefficients a, b, and c. *Since the Quadratic Formula involves the square root we can have a variety of different solutions; from rational to radical to complex.

The derivation of the Quadratic Formula

Using the quadratic formula to solve equations…. Make sure your equation is in the form, ax²+bx+c=0, if it isn’t then use algebra to put it in this form. Identify the coefficients a, b, and c. Plug them into the equation and simplify the result. These are the two solutions, they are radical solutions

Examples… Solve:

Examples… Solve: *The solutions are approximately and if rounded to 4 decimal places.

Solving Quadratic Inequalities We can use our factoring and solution methods to determine when a quadratic has positive and negative output values. First find the zeros for the quadratic. Place them on a number line and test values on each side of the zeros to determine the sign of the region. + means the region is positive - means the regions is negative List all of the regions that satisfy the inequality in interval notation. Try x=-2 + Try x=0 - Try x=3 + 2 Solution: (-∞,-1)U(2,∞)

Examples…. Solve the inequality : Try x=-3 + Try x=0 - Try x= Solution: (-2,5)

Examples…. Solve the inequality : Try x=-6 + Try x=-2 - Try x= Solution: (-∞,-5]U[-1,∞)

Inverses-3-7 topic p

Finding an inverse for a function… For a function f(x)=rule, put in y=rule form. Swap y with x and vice versa. Solve for y. The inverse is the function y=new rule and is denoted:

Examples…. 1.Find the inverse for f(x)=2x+3. Swap the x’s and y’s! Solve for y!

Examples…. 2. Find the inverse of Swap the x’s and y’s! Solve for y!