Solving Equations Algebraically and Graphically

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.

Diophantis Father of Algebra First to use abbreviated word forms in English

... his boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/12th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son. So he married at the age of 26 and had a son who died at the age of 42, four years before Diophantus himself died aged 84. Based on this information we have given him a life span of 84 years.

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

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.

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.)

Solve:

Solve: No solution

Intercepts and Solutions A point at which the graph of an equation meets the x-axis is called an x-intercept. 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 y-intercept. We find it be replacing x with 0 and solving for y.

Find the x- and y- intercepts of the graph of each equation. 2x +3y = 6

y = x2 + x - 6

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.

Use a graphing utility to approximate the solutions of x3 + 4x + 1 = 0

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.

Find the points of intersection of the graphs of: x - 2y = 1 and 3x - y = 7

Find the points of intersection of the graphs of: y = x2 + 2x - 8 and y = x3 + x2 -6x + 2 (-3.31863, -3.62396).

Solving Polynomial Equations Algebraically Quadratic Equations Factoring Square root principle Completing the square Quadratic Formula

Solve by factoring: x2 + 7x +12 = 0

The Illegal Move The illegal move is used to factor a quadratic equation when the leading coefficient (a) is not 1.

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.

Step 2: Now, factor this new trinomial.

Step 3: Undo the “Illegal Move” by dividing the original leading coefficient (a).

Step 4: Reduce the fractions.

Step 5: Clear the fractions by moving the denominator in front of the x.

Solve by factoring: 2x2 + 3x = -1

The Square Root Principle Get “squared” part by itself. Take the square root of both sides. Solve both equations for x.

Solve using the square root principle: 16x2 = 25

Solve using the square root principle: (x - 4)2 + 3 = 12

Completing the Square Step 1: Move c to the other side. Step 2: Add to both sides. Step 3: Factor your perfect square trinomial. Step 4: Complete using the Square Root Principle.

Solve by completing the square: x2 + 4x = 5

Solve using the quadratic formula: 3x2 - x - 5 = 0

Solving Quadratic Type Equations Write in quadratic form. Factor. Solve. No solution

Please solve

Other types of equations WARNING!!!!!!! The following methods can produce extraneous roots. Therefore all alleged solutions should be checked in the original equation.

Solving An Equation With Rational Exponents

Solving an Equation Involving Fractions

Solving An Equation Involving Absolute Value