This document provides an overview of different methods for solving quadratic equations, including factoring, graphing, using the quadratic formula, and more. It begins by defining the general and standard forms of quadratic equations. It then explains how to write equations in standard form and discusses concepts like the vertex, completing the square, determining zeroes/roots, and the discriminant. Finally, it reviews several methods that can be used to find roots of quadratic and higher-order polynomial equations, such as factoring, graphing, the quadratic formula, synthetic division, and the remainder/factor theorems.

1. Solving Quadratic Equations by Factoring and Graphing By: Megan Littlewood

2. Equations of Quadratics There are two basic forms of quadratic equations: General form: y= ax 2 +bx+c Standard form: y=a(x-h) 2 +c (or k) Example of standard form: Y=(x+6) 2 +2 x- coordinate but with the opposite sign y-coordinate of the vertex Vertex = (-6,2)

3. Writing Equations in Standard Form We’ll start with an example: Find the equation when the vertex is (3,-1) and x-intercepts of 2 and 4 Y= a(x+h) 2 +k Y=a(x-3) 2 -1 0=a(2-3) 2 -1 0=a(-1) 2 -1 1=a(-1) 2 a=1 So the equation of the line is y= (x-3) 2 -1 Plug the vertex coordinates in for h and k, remember to flip the sign for the h value! Let y=0 and x=2 to solve for a Next we’ll learn some other ways to solve quadratic equations!

4. Completing the Square (Going from general form to standard form) Example: Put y=x 2 +10x+23 into standard form Section off the ‘x’ terms Y=(x 2 +10x)+23 Add a blank at the end of the bracket Y=(x 2 +10x+___)+23 Subtract this number from the ‘c’ value 23-25 Y=(x 2 +10x+25)+23-25 Factor the brackets and simplify Y=(x+5) 2 -2 Hooray! You have now completed the square! 10/2=5 5 2 =25 Plug this number into the blank

5. Vertex Formula From the general formula we can conclude that Y=-b/2a X=4ac-b 2 /4a Therefore the vertex formula is: (-b/2a, 4ac-b 2 /4a) Example: y=-4x 2 +12x +5 a=-4 v= -12 b=12 2(-4) c=5 , 4(-4)(5)-(12) 2 4(-4) V= (3/2, 14)

6. Determining the Zeroes of a Quadratic The zeroes are also called: Roots - Real roots Real solutions - X-intercepts There are three possibilities for types of roots Two distinct roots—line touches x-axis twice One root—line touches x-axis once No real roots—line does not touch the x-axis Example: If after factoring the solution is (x-3) and (x+2), then the zeroes are 3 and -2. Notice that the zeroes are the x value which will make the solution equal zero

7. The Quadratic Formula First of all, the equation must be in the form: y=ax 2 +bx+c Also make sure that the quadratic is in descending powers of x Note that there will probably be two answers The Formula: x=-b+- b 2 -4ac 2a Example:2x 2 -5x+2=0 X=-(-5)+- (-5) 2 -4(2)(2) 2(2) X=5+- 9 4 x=2 and x=1/2 These are the zeroes of the equation **For the discriminant remember that no value of x can ever make something in the denominator equal zero

8. The Square Root Principle You can use the square root principle when a number in the equation has a perfect square Example: Solve: x 2 -14=155 x 2 = 169 x=+-13 **Note: Make sure you account for both the positive and negative answer Example 2: x 2 -21=-3 x 2 -18=0 x 2 = 18 x=+-3 2

9. The Discriminant The radicand from the quadratic formula is called the discriminant b 2 -4ac We can use the discriminant to determine the nature of the roots There are three options: b 2 -4ac>0 b 2 -4ac=0 b 2 -4ac<0 There are two real and distinct roots There are two equal roots There are no real roots **Note: The equation must be in the general form, in descending powers of x

10. The Remainder Theorem The Remainder Theorem is used when a polynomial has a degree higher than 2 Plug the x value in for x in the equation and solve, finding the remainder Example: Find the remainder when (x 3 -x+28) is divided by (x+3) x=-3 (-3) 3 -(-3)+28 Remainder = 4

11. The Factor Theorem The Integral Zero Theorem- A factor is a value that divides another evenly, the remainder will always be zero Use the value of the constant to find the factor(s) that will equal zero Example: Find the factors of x 3 +2x 2 -5x-6 -6 +- 1,2,3,6—Guess and check with calculator, continue until you reach zero F(-1)=0—Now to find the other factors, divide the original equation by (x+1) Since the remainder is zero after the division, there are no more factors

12. Rational Zero Theorem Use this Theorem when the leading coefficient is not 1 x= b/a b is a factor of the constant a is a factor of the first term Example: Find the factors of 6x 3 +13x 2 +x-2 -2 6 +- 1,2 +- 1,2,3,6 F(-1/2)=0 (Therefore (2x+1) is a factor) (2x+1)(6x 2 +10x-4) (2x+1)(3x-1)(x+2) Factor by sum and product:

13. Synthetic Division Is used as a quicker way to divide, easier than using long division Example: Divide 2x 3 -4x 2 +3x-6 by x+2 2 - Note: Always subtract on the bottom For the top, use the opposite sign of what x equals 2 -4 3 -6 2 -8 19 -44 The factors are: (x+2)(2x 2 -8x+19)-44 (x+2)2(x-2) 2 +11-44 2(x+2)(x-2) 2 -33 x=-2 and x=2 and x=2 The zeroes are:

14. Solving by Graphing On your graphing calculator, follow these steps to answer quadratic related questions (in a way that does not involve a pencil!) Example: Y=-x 2 +3 Y=-x+2 Hit: CALC INTERSECT (Do twice) Scroll the curser to the point of intersection (as close as you can get). Hit guess and record the answer, repeat a second time on the second point of intersection

15. Review There are several methods to finding the roots: Graphing (doesn’t always give exact values of real roots Factoring by sum and product (not all can be factored) Completing the Square Quadratic Formula Integral/Rational Zero Theorem If polynomial has a degree higher than 2: Remainder/Factor Theorem Synthetic Division

16. THE END Math 20 Pure January 2011