intermediate algebra chapter 8 quadratic equations
TRANSCRIPT
Willa Cather –U.S. novelist
• “Art, it seems to me, should simplify. That indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the whole – so that all one has suppressed and cut away is there to the reader’s consciousness as much as if it were in type on the page.
Solving Quadratic Equation #1
• Factoring• Use zero Factor Theorem• Set = to 0 and factor• Set each factor equal to zero• Solve• Check
Solving Quadratic Equation #2
• Graphing
• Solve for y
• Graph and look for x intercepts
• Can not give exact answers
• Can not do complex roots.
Solving Quadratic Equations #3Square Root Property
• For any real number c
2if x c then
x c or x c
x c
Procedure
• 1. Use LCD and remove fractions
• 2. Isolate the squared term
• 3. Use the square root property
• 4. Determine two roots
• 5. Simplify if needed
Completing the square informal
• Make one side of the equation a perfect square and the other side a constant.
• Then solve by methods previously used.
Procedure: Completing the Square• 1. If necessary, divide so leading
coefficient of squared variable is 1.
• 2. Write equation in form
• 3. Complete the square by adding the square of half of the linear coefficient to both sides.
• 4. Use square root property
• 5. Simplify
2x bx k
Mary Kay Ash
• “Aerodynamically, the bumble bee shouldn’t be able to fly, but the bumble bee doesn’t know it so it goes flying anyway.”
Quadratic Formula
• For all a,b, and c that are real numbers and a is not equal to zero
23 8 7 0
4 5
3 3
x x
x i
2 4
2
b b acx
a
Pearl S. Buck
• “All things are possible until they are proved impossible and even the impossible may only be so, as of now.”
Methods for solving quadratic equations.
• 1. Factoring
• 2. Square Root Principle
• 3. Completing the Square
• 4. Quadratic Formula
Discriminant
• Negative – complex conjugates• Zero – one rational solution (double
root)• Positive
– Perfect square – 2 rational solutions– Not perfect square – 2 irrational
solutions
2 4b ac
Harry Truman – American President
• “A pessimist is one who makes difficulties of his opportunities and an optimist is one who makes opportunities of his difficulties.”
Orison Swett Marden
• “All who have accomplished great things have had a great aim, have fixed their gaze on a goal which was high, one which sometimes seemed impossible.”
Vertex
• The point on a parabola that represents the absolute minimum or absolute maximum – otherwise known as the turning point.
• y coordinate determines the range.
• (x,y)
Axis of symmetry
• The vertical line that goes through the vertex of the parabola.
• Equation is x = constant
Graphing Quadratic• 1. Determine if opens up or down• 2. Determine vertex• 3. Determine equation of axis of
symmetry• 4. Determine y intercept• 5. Determine point symmetric to y
intercept• 6. Determine x intercepts• 7. Graph