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Slide 1 ___________________________________ ___________________________________ P.4 Solving Equations Algebraically and Graphically ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 2 • An ancient Egyptian papyrus, discovered in 1858, contains one of the earliest examples of mathematical writing in existence. • The papyrus itself dates back to around 1650 B.C., but it is actually a copy of writings form two centuries earlier. • The algebraic expressions on the papytrus were written in words. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 3 ___________________________________ Diophantis • Father of Algebra • First to use abbreviated word forms in English ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 4 • ... his boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/ th more, and his son was born 5 years 12 later; the son lived to half his father's age, and the father died 4 years after the son. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 5 Equations and Solutions of Equations An equation in x is a statement that variable expressions are equal. A solution of an equation is a number r, such that when x is replaced by r, the resulting equation is a true statement ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 6 ___________________________________ The solution set of an equation is the set of all solutions of the equations. To solve an equation means to find its solution set. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 7 ___________________________________ Types of Equations Identity - every real number in the domain of the variable is a solution. Conditional - only some of the numbers in the domain of the variable are solutions. (These are the types we have to solve.) ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 8 ___________________________________ Solve: x 3x 2 3 4 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 9 ___________________________________ Solve: 1 3 6x x 2 x 2 x2 4 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 10 ___________________________________ Intercepts and Solutions A point at which the graph of an equation meets the x-axis is called an xintercept. We find it be replacing y with 0 and solving for x. A point at which the graph of an equation meets the y-axis is called a yintercept. We find it be replacing x with 0 and solving for y. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 11 ___________________________________ Find the x- and y- intercepts of the graph of each equation. 2x +3y = 6 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 12 • y = x2 + x - 6 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 13 Finding Solutions Graphically Write the equation in general form, f(x) = 0 Use a graphing utility to graph the function y = f(x). Be sure the viewing window shows all the relevant features of the graph. Use the zero or root feature or the zoom and trace features of the graphing utility to approximate the x-intercept of the graph of f. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 14 ___________________________________ Use a graphing utility to approximate the solutions of x3 + 4x + 1 = 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 15 Points of Intersection of Two Graphs Points at which two graphs meet are called points of intersection. Their corresponding ordered pairs are solutions to both of the equations. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 16 ___________________________________ Find the points of intersection of the graphs of: x - 2y = 1 and 3x - y = 7 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 17 ___________________________________ Find the points of intersection of the graphs of: y = x2 + 2x - 8 and y = x3 + x2 -6x + 2 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 18 Solving Polynomial Equations Algebraically Quadratic Equations Factoring Square root principle Completing the square Quadratic Formula ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 19 ___________________________________ Solve by factoring: ___________________________________ x2 + 7x +12 = 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 20 ___________________________________ The Illegal Move ax 2 bx c 0 ___________________________________ ___________________________________ The illegal move is used to factor a quadratic equation when the leading coefficient (a) is not 1. ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 21 6x2 7x 3 0 Step 1: Multiply the leading coefficient (a) by c and form a new trinomial where a is now 1, b is the same, and c is now ac. x 2 7 x 18 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 22 x 2 7 x 18 0 Step 2: Now, factor this new trinomial. x 9x 2 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 23 x 9x 2 0 Step 3: Undo the “Illegal Move” by dividing the original leading coefficient (a). 9 2 x x 0 6 6 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 24 9 2 x x 0 6 6 Step 4: Reduce the fractions. 3 1 x x 0 2 3 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 25 3 1 x x 0 2 3 ___________________________________ ___________________________________ Step 5: Clear the fractions by moving the denominator in front of the x. ___________________________________ 2x 33x 1 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 26 ___________________________________ Solve by factoring: 2x2 ___________________________________ + 3x = -1 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 27 ___________________________________ The Square Root Principle • Get “squared” part by itself. • Take the square root of both sides. • Solve both equations for x. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 28 ___________________________________ Solve using the square root principle: ___________________________________ 16x2 = 25 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 29 ___________________________________ Solve using the square root principle: (x - 4)2 ___________________________________ + 3 = 12 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 30 ___________________________________ Completing the Square • Step 1: Move c to the other side. • Step 2: Add b2 to both sides. • Step 3: Factor your perfect square trinomial. b 2 b2 ___________________________________ 2 x c 2 4 • Step 4: Complete using the Square Root Principle. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 31 ___________________________________ Solve by completing the square: ___________________________________ x2 + 4x = 5 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 32 ___________________________________ Solve using the quadratic formula: 3x2 ___________________________________ -x-5=0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 33 ___________________________________ Solving Quadratic Type Equations x 4 5x 2 6 0 •Write in quadratic form. x 2 2 •Factor. x 2 ___________________________________ 5 x2 6 0 2 x2 3 0 ___________________________________ •Solve. ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 34 x 4 8 x 2 15 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 35 ___________________________________ Please solve x 9x 3 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 36 x3 x 2 4 x 4 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 37 ___________________________________ Other types of equations WARNING!!!!!!! The following methods can produce extraneous roots. Therefore all alleged solutions should be checked in the original equation. ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 38 Solving An Equation With Rational Exponents 3 2 ___________________________________ ___________________________________ x 27 0 ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 39 Solving an Equation Involving Fractions x 6 2 x 1 x ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ Slide 40 Solving An Equation Involving Absolute Value x2 6 x ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________ ___________________________________