Quadratics

Quadratics

1. General Form of a Quadratic Equation

   • f(x)= ax^2+bx+c
     • Example: f(x)= 2x^2+x
       • a=2, b=1,c=0

2. General Form from Factored Form

   • Example: f(x)=(3x-6)(x+5)
     • FOIL
       • f(x)= 3x^2+15x-6x-30
         • Simplify and Combine Like Terms
           • ANSWER: 3x^2+9x-30

3. Standard Form of a Quadratic Equation

   • f(x)=a(x-h)^2+k
     • can be solved by completing the square
   • Also known as Vertex Form
     • Vertex Form of a Parabola
       • y=a(x-h)^2+k
         • Graph is skinnier if a>1, Graph is wider if 0< a<1
           • If the "a" value is positive the graph opens up, if the "a" value is negative the graph opens down
             • h= horizontal shift left or right
               • k= vertical shift up or down
                 • minus=right, plus=left
                 • Vertex= (h,k)

4. Finding the quadratic function when the graph is given

   • Find Vertex
     • Find one other point on the graph
       • Plug in vertex
         • Plug in point and solve for "a"
           • Write equation

5. Solving Quadratic Equations using the square root property

   • Isolate the squared variable expression
     • Use the square root property to undo the square
       • Rewrite as two equations and solve
         • Check answers in the original equation
           • Be sure the answers are within the domain of the model and that they make sense in the context of the problem
             • Example: 4(x+7)^2=400
               • Divide by 4 to both sides
                 • (x+7)^2=100
                   • Take the square root of both sides
                     • x+7= positive or negative 10
                       • x+7=-10
                         • subtract 7 to both sides
                           • x=3
                       • x+7=10
                         • subtract 7 to both sides
                           • x=-17
       • Use + and - on the side away from the variable

6. Solving Quadratic Equations by Completing the Square

   • Isolate the variable terms on one side of the equation
     • If the coefficient of x^2 is not 1, divide both sides of the equation by the coefficient of x^2
       • Take half the coefficient of x, then square it
         • Add that number to both sides of the equation
           • Factor the quadratic into the square of a binomial
             • solve using the square root property
               • check answers in the original equation
                 • Example: a^2+2ab+b^2
                   • x^2+6x+9
                     • (6/2)^2=9
                       • ANSWER: (x+3)^2
                 • Example:a^2-2ab+b^2
                   • x^2-10x+25
                     • (-10/2)^2=25
                       • (x-5)^2

7. Solving Using the quadratic formula

   • -b + or - the square root of b^2 -4ac all over 2a
     • Set the quadratic equal to zero
       • Put the quadratic into standard form
         • Substitute the values of a,b, and c into the quadratic formula
           • Simplify the quadratic formula
             • Check answers in the original equation
               • Example: 3x^2+4x-3=0
                 • -4+ or - the square root of 4^2-4(3)(-3)/2(3)
                   • -4+ 2 square root of 13
                   • -4-2 square root of 13
                   • ANSWER: -2+ or - the square root of 13/(3)

8. Graph of a Quadratic

   • is called a Parabola
     • Vertex
       • The turning point of the graph
         • The vertex is the MINIMUM if the parabola is open upward
           • The vertex is the MAXIMUM if the parabola is open downward
             • AXIS OF SYMMETRY
               • The vertical line that goes through the vertex

9. MINIMUM

   • Lowest point on the graph
     • MAXIMUM
       • Highest point on the graph
         • if a<0, then the maximum value is f(-b/2a)
     • if a> 0, then the minimum value is f(-b/2a)