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Lecture 4 solving other types of equations
1.
PROCEDURE FOR SOLVING EQUATIONS
BY FACTORING Step 1 Make one side zero. Move all nonzero terms in the equation to one side (say the left side), so that the other side (right side) is 0. Step 2 Factor the left side. Step 3 Use the zero-product property. Set each factor in Step 2 equal to 0, and then solve the resulting equations. Step 4 Check your solutions. © 2010 Pearson Education, Inc. All rights reserved 1
2.
EXAMPLE 2 Solving
an Equation by Factoring Solve by factoring: 4 2 9x x © 2010 Pearson Education, Inc. All rights reserved 2
3.
EXAMPLE 3 Solving
an Equation by Factoring Solve by factoring: 3 2 2 2x x x © 2010 Pearson Education, Inc. All rights reserved 3
4.
EXAMPLE 4 Solving
a Rational Equation Solve: 1 1 1 6 1x x © 2010 Pearson Education, Inc. All rights reserved 4
5.
EXAMPLE 5 Solving a
Rational Equation with an Extraneous Solution Solve: © 2010 Pearson Education, Inc. All rights reserved 5
6.
SOLVING EQUATIONS CONTAINING SQUARE
ROOTS Step 1 Isolate one radical to one side of the equation. Step 2 Square both sides of the equation in Step 1 and simplify. Step 3 If the equation in Step 2 contains a radical, repeat Steps 1 and 2 to get an equation that is free of radicals. Step 5 Check the solutions in the original equation. Step 4 Solve the equation obtained in Steps 1 - 3. © 2010 Pearson Education, Inc. All rights reserved 6
7.
EXAMPLE 6 Solving
Equations Involving Radicals Solve: 3 6x x x © 2010 Pearson Education, Inc. All rights reserved 7
8.
EXAMPLE 7 Solving
an Equations Involving a Radical Solve: 2 1 1x x © 2010 Pearson Education, Inc. All rights reserved 8
9.
EXAMPLE 8 Solving
an Equation Involving Two Radicals Solve: 2 1 1 1x x © 2010 Pearson Education, Inc. All rights reserved 9
10.
SOLVING EQUATIONS OF
THE FORM um/n = k © 2010 Pearson Education, Inc. All rights reserved 10 Let m and n be positive integers, k a real number, and in lowest terms. Then if m n
11.
EXAMPLE 9 Solving
Equations with Rational Exponents Solve. Solution © 2010 Pearson Education, Inc. All rights reserved 11
12.
An equation in
a variable x is quadratic in form if it can be written as EQUATIONS THAT ARE QUADRATIC IN FORM 2 0 0 ,au bu c a where u is an expression in the variable x . We solve the equation au2 + bu + c = 0 for u. Then the solutions of the original equation can be obtained by replacing u by the expression in x that u represents. © 2010 Pearson Education, Inc. All rights reserved 12
13.
EXAMPLE 10 Solving an
Equation That Is Quadratic in Form Solve: Solution Let u = x1/3, then u2 = (x1/3)2 = x2/3. © 2010 Pearson Education, Inc. All rights reserved 13
14.
EXAMPLE 11 Solving an
Equation That Is Quadratic in Form Solve: 2 1 1 6 8 0x x x x © 2010 Pearson Education, Inc. All rights reserved 14
15.
EXAMPLE 12 Investigating
Space Travel Your sister is 5 years older than you are. She decides she has had enough of Earth and needs a vacation. She takes a trip to the Omega-One star system. Her trip to Omega-One and back in a spacecraft traveling at an average speed v took 15 years, according to the clock and calendar on the spacecraft. But on landing back on Earth, she discovers that her voyage took 25 years, according to the time on Earth. © 2010 Pearson Education, Inc. All rights reserved 15
16.
EXAMPLE 12 Investigating
Space Travel This means that, although you were 5 years younger than your sister before her vacation, you are 5 years older than her after her vacation! Use the time-dilation equation to calculate the speed of the spacecraft. 2 0 2 1 v t t c © 2010 Pearson Education, Inc. All rights reserved 16
17.
EXAMPLE 12 Investigating
Space Travel Substitute t0 = 15 (moving-frame time) and t = 25 (fixed-frame time) to obtain 2 2 2 2 2 2 1 1 3 1 5 9 1 25 5 25 v c v c v c Solution 2 2 2 9 1 25 16 25 v c v c 4 5 4 0.8 5 v c v c c So the spacecraft was moving at 80% (0.8c) the speed of light. © 2010 Pearson Education, Inc. All rights reserved 17
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