Honors Topics

 You learned how to factor the difference of two perfect squares:  Example:  But what if the quadratic is ? You learned that it was not factorable. It is factorable, but with complex factors.  How does it work? Let’s solve an equation. If you solve the equation, you get solutions x =1,-1. If you solve the equation, you get the solutions x = i,-i. Since a quadratic factors into (x – zero)(x-zero):

 Factor each quadratic expression.

 You must always simplify powers of the imaginary unit, i. You learned the properties: i² = -1 and But how would you simplify ? You need to find out how many pairs/sets of imaginary units, i, you have and simplify them. Since every i² = -1,

 Simplify each imaginary number.

 Recall that you must rationalize the denominator of a fraction. You cannot leave a radical on the bottom of a fraction. You also cannot leave a complex number on the bottom of a fraction.  Conjugates: 4-6i, 4+6i ½ + i, ½ - I 7i, -7i Divide by a Complex Number a + bi Multiply numerator and denominator by the conjugate of the bottom, a – bi

 Divide.

 Solve a quadratic equation in standard form with leading coefficient of 1  Factored form is (x-m)(x-n) = 0 Therefore x = m, x = n are the roots. m + n = = -3, which is -b m · n = (-2)(-1) = 2, which is c If a = 1 and, then: Sum of the roots: m + n = -b Product of the roots: m · n = c or Note: if a≠1, sum = -b/a and product = c/a

 Given the zeros of the quadratic, write an equation in standard form.  x = 5, -4 {-6, 0} x = 5±3i  5+-4 = = -6 (5+3i)+(5-3i) = 10  So -b = 1 So –b = -6 So –b = 10  b = -1 b = 6 b = -10  5(-4) = -20 (-6)(0) = 0 (5+3i)(5-3i)= 34  So c = -20 So c = 0 So c = 34

 Recall the standard form of quadratic equation:.  The discriminant,, of the quadratic indicates the nature of the roots.  2 real roots 1 real root 2 complex roots  Q: How would you find the value of a, b, or c in a quadratic equation given that there are 2 real roots? 1 real root? 2 complex roots?  A: Set up an equation or inequality and solve!

 Given, find the value(s) of c if there is/are:  2 real roots 1 real root 2 complex roots *C is less than 1 *C is equal to1 *C is greater than 1

 You learned how to solve quadratic equations using different methods and you learned how to solve quadratic inequalities by graphing. Now let’s look at solving inequalities algebraically. Solving Quadratic Inequalities Using Algebra 1.Write the inequality as an equation. 2. Solve the quadratic equation by factoring. 3.Graph each solution on the same number line. 4. Test a number in all three sections of the number line by substituting it into the original inequality. If the inequality is True, each number in that section is a solution. 5. Write your solutions as an inequality. If m and n are the zeros for the quadratic equation, the solutions will be: solution 1: m n

 Solve the inequality  Write it as an equation and find the roots  Graph the roots -2, 2 on the number line  Use a closed circle since it is ≤  Check values in each section  x = -2, 0, and 3  Solution: -1 ≤ x ≤ 2  Note: if the inequality is, then the False sections would now be true. The solution would be x 2.