Mastering Quadratic Equations: Methods and Techniques
Explore various methods to solve quadratic equations, including factoring, graphing, completing the square, and using the quadratic formula. Learn how to simplify the process and find solutions effectively.
Mastering Quadratic Equations: Methods and Techniques
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Presentation Transcript
Intermediate AlgebraChapter 11 • Quadratic Equations
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.
Intermediate Algebra 11.1 • Special Methods
Def: Quadratic Function • General Form • a,b,c,are real numbers and a not equal 0
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
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
Dorothy Broude • “Act as if it were impossible to fail.”
Intermediate Algebra 11.1 Gay • Completing • the • Square
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
Objective: • Solve quadratic equations using the technique of completing the square.
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.”
Intermediate Algebra 11.2 • The • Quadratic • Formula
Objective of “A” students • Derive • the • Quadratic Formula.
Quadratic Formula • For all a,b, and c that are real numbers and a is not equal to zero
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
CalculatorPrograms • ALGEBRAQUADRATIC • QUADB • ALG2 • QUADRATIC
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.”
Intermediate Algebra 11.4 • Quadratic Inequalities
Intermediate Algebra 11.5-11.6 • Quadratic Functions
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
Objective • Graph, determine domain, range, y intercept, x intercept
Parabola with vertex (h,k) • Standard Form
Find Vertex • x coordinate is • y coordinate is
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
