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  • 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