Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero

Size: px

Start display at page:

Download "Equations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero"

Roger Jeffery Goodwin
6 years ago
Views:

1 Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve, excluding values that cause the denominator to equal zero Check solutions using the original equation Example ( x 1)(x + 5) ()() x + x 5 x + x x + 5 x Cross Multiply 1 Remove parentheses Write in standard form. x + x b ± b ac x a b c a ± ()( ) () Substitute the values of a, b, and c into the quadratic formula ± + 7 ± 81 ± 1

2 Example x 5 ± x + x 1 x 1 x x + x 6 x x Simplify Example x 5 x x ( ) + 5 ( ) 1 ( ) ( ) {, }. + x 1 5 Example Using the TI-8 x 5 Y1 LHS - RHS + x 1

3 Example Using the TI-8 x 5 Y1 LHS Y RHS + x 1 Example Using the TI-8 x 5 + x 1 Solving Radical Equations When the variable in an equation occurs in a radicand, the equation is called a radical equation. 5x 1 a + 7 6c+ 8+ c 1 Solving a Radical Equation Containing One Radical Step 1: Isolate the radical. That is, get the radical by itself on one side of the equation and everything else on the other side. Step : Raise both sides of the equation to the power of the index. This will eliminate the radical from the equation. Step : Solve the equation that results. Step : Check your answer. When solving radical equations containing an even index, apparent solutions that are not solutions to the original equation are called extraneous solutions.

4 Example 5x 1 ( ) 5x 1 Square each side 5x 1 5x 1 x Simplify and solve for x Check: 5x 1 5() 1 1 Example y y+ y+ y+ Isolate the radical ( y ) ( y ) + + Square each side y + 6y+ y+ FOIL each side y + y Simplify and set equal to y( y + ) Factor the quadratic equation y and y Solve for y 11 Example y y+ y and y Check y y+ ( ) + y y+ ( ) False is an extraneous root. The only solution is y 1

5 Solving Radical Equations Solving a Radical Equation Containing Two Radicals Step 1: Isolate the radical. That is, get the radical by itself on one side of the equation and everything else on the other side. Step : Raise both sides of the equation to the power of the index. This will eliminate one radical or both radicals from the equation. Step : If a radical remains in the equation, then follow the steps for solving a radical equation containing one radical. Otherwise, solve the equation that results. Step : Check your answer. When solving radical equations containing an even index, apparent solutions that are not solutions to the original equation are called extraneous solutions. 1 Example y+ 1 y y+ 1 + y Isolate one of the radicals ( + 1) ( + ) y y Square each side y y + y FOIL each side y 6 y Isolate the remaining variable ( ) ( 6 ) y y Square each side y 16y+ 16 6( y ) Simplify Example y+ 1 y y 16y+ 16 6( y ) y 16y+ 16 6y 1 Simplify y 5y+ 16 Simplify ( y y ) 1 + Factor out a ( y 5)( y 8) Factor the trinomial y 5 y 8 Set each factor equal to y 5 and y 8 Solve for 5

6 Example x + x + x + x + x x x 8x+ 16 x+ 7+ x x+ + 1 ( x 7)( x ) 1 x, Solution is { } Equations with Rational Exponents uv, are expressions mn, are positive integers mn If m is odd, then u v u v nm mn If m is even and v>, then u v u ± v nm Example ( x + 5) x + 5± x + 5± 7 x 5± 7 x 5 7 x 5+ 7 {,} 6

7 Example ( x ) 8 x 8 x x 7 { 7} Equations in Quadratic Form The idea is to get the equation in the form of a quadratic, which we know how to solve We usually do this by substitution After solving, again check for extraneous solutions Examples Equation Substitution x 1x + t y y + y 1/ 15 t y 1/ x / x 1/ t x 1/ (1+x) + (1+x) - t x + 1 7

8 Solving Quadratic Type Equations Step 1 Write in Standard Form ax + bx + c if needed Step Substitute a variable in for the expression that follows b in the second term Step Solve the quadratic equation created in step Step Find the value of the variable from the original equation Step 5 Check your solutions Example y + y Substitute a variable in for the Let t y expression that follows b in the second term t + t Factor ( t )( t ) + 1 Solve Rewrite in terms of original variable t,1 y,1 Solve y ±, i ± 1 Example y + y Check y i ( i) ( i) + ( ) y i ( i) ( i) + ( )

9 Example y + y Check y 1 y 1 ( ) ( ) () () { i i } The solution is,, 1,1 Example ( x x) ( x x) Let t x + x t 8t+ 16 ( ) t t x + x x + x ( x )( x ) + 1 x,1 Example Check x x 1 ( x x) ( x x) (( ) ( )) ( ) ( ) () 8() ( ) (() ()) (() ()) () 8() The solution is,1 { }

10 Practice s 1s + Let t s t 1t+ ( t )( t ) t 1 t t 1, s 1, s + 1s s ± 1, ± Practice Check s + 1s s 1 ( ) + ( ) s 1 () + () Practice Check s ( ) ( ) { } Solution set, 1,1, s + 1s s ( ) ( )

11 Practice s + 1s Problem Solving Read the problem carefully Pick out key words and phrases and determine their equivalent mathematical meaning Replace any unknowns with a variable Put it all together in an algebraic expression. Translation into algebra Addition: sum, plus, add to, more than, increased by, total Subtraction: difference of, minus, subtracted from, less than, decreased by, less Multiplication: product, times, multiply, twice, of Division: quotient, divide, into, ratio 11

12 Steps to Solving Problems Understand the problem Devise a plan, translate Carry out the plan, solve Look back, check and interpret Example Three times the difference of a number and is 8 more than that number. Find the number. Three times the difference of x and is 8 more than x 8 + x ( x ) ( x ) 8+ x x 1 8+ x x x 1 Three times the difference of 1 and, that is the same as 8 more than 1, so this does check Solution: The number is 1 1

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!!

Never leave a NEGATIVE EXPONENT or a ZERO EXPONENT in an answer in simplest form!!!!! 1 ICM Unit 0 Algebra Rules Lesson 1 Rules of Exponents RULE EXAMPLE EXPLANANTION a m a n = a m+n A) x x 6 = B) x 4 y 8 x 3 yz = When multiplying with like bases, keep the base and add the exponents. a