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Section 4 - Solving Equations Algebraically and Graphically

Page history last edited by PBworks 16 years, 3 months ago

Section P.4: Solving Equations Algebraically and Graphically

 

 

Conditional- an equation that is true for some (or no) real numbers

Ex: eq=x+1=2

      eq=x=1 <--There is only 1 answer!

 

Identity- an equation that is true for all numbers

Ex: eq=2x+4=2(x+2) <-- X can equal anything because of the distributive property.

 

                     Ex. Find the Least Common Denominator (LCD)

 

eq=frac{x}{3} + frac{3x}{4} = 2                                                                                eq=frac{1}{x-2}=frac{3}{x+2}-frac{6x}{x^2-4}

 

eq=\frac{x}{3} + \frac{3x}{4} = \frac{2}{1}                                                                               eq=frac{1}{x-2}=frac{3}{x+2}-frac{6x}{(x+2)(x-2)} <--the LCD is (x+2)(x-2)

 

eq=12 (\frac{x}{3} + \frac{3x}{4}) = 12 (\frac{2}{1}) <--the LCD is 12, the lowest number         eq=x+2=3(x-2)-6x

                                          with both 3 and 4 as factors

eq=4x + 9x = 24                                                                              eq=x+2=3x-6-6x

 

eq=13x = 24                                                                                      eq=8=-4x

 

eq=x = \frac{24}{13}                                                                                         eq=x=-2      BUT this leads a denominator of 0,        

                                                                                                                      so -2 is extraneous...NO  SOLUTION!

 

                                        ***Check Answers for: ***

                                          Variables in denominators

                                          Equations with eq=sqrt{number}

 

Ex. Find x and y intercepts of eq=2x + 3y = 5

 

eq=3y = 5                     eq=2x = 5         

eq=y = frac{5}{3}                       eq=x = frac{5}{2}

 

 

 

 

 

 Ex. Use  graphing calculator to approximate the solutions of eq=2x^3-3x^2+2=0

      (Go to Y= and insert the equation, press graph) The graph will look like this...

 

 

 

 On your calculator go to the CALC function, scroll down to "zeroes" and find the zero(es)!

                eq=x\approx-.678

 

 

Solving quadratics  eq=ax^2 + bx + c

1. Factor (if you can)

Otherwise...

2. Completing the square

eq=x^2 + 6x - 2 = 0 

Add 2 to both sides

eq=x^2 + 6x = 2

Take half of 6 (b) and square it. Then add it to both sides.

eq=x^2 + 6x + 9 = 2 + 9

Now you can factor

eq=(x + 3)(x + 3) = 11

Rewrite it

eq=(x + 3)^2 = 11

3. Square Root (This goes on its own, and also to finish the above equation)

eq=\sqrt{(x + 3)^2} = \sqrt{11}

The square root and the square cancel

eq=(x + 3) = \pm\sqrt{11}

Solve

eq=x = -3 \pm\sqrt{11}

4. Quadratic Formula

eq=x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a }

 

 

Ex. eq=x^4 - 3x^2 + 2 = 0 

(Quadratics to the 4th and 2nd powers work just like the other ones)

eq=(x^2 - 2)(x^2 - 1) = 0

eq=x^2 = 2                             eq=x^2 = 1

eq=\sqrt{x^2} = \sqrt{2}                       eq=\sqrt{x^2} = \sqrt{1}

eq=x = \pm\sqrt{2}                        eq=x = \pm1

 

 

 

Ex. eq=2x^3 - 6x^2 - 6x +18 = 0     Solve by grouping

Divide by 2

eq=x^3 - 3x^2 - 3x +9 = 0

Pull eq=x^2  out of the first two terms and -3 out of the second two

eq=x^2 (x - 3) - 3(x - 3) = 0

Combine the outside terms and multiply by the parenthesis

(Reverse distributive property)

eq=(x^2 - 3)(x - 3) = 0

eq=x - 3 = 0                      eq=x^2 - 3 = 0         

                                   eq=x^2 = 3

eq=x = 3             and         eq=x = \pm3  

 

 

 

Ex. eq=(x + 1)^\frac{2}{3} = 4 

Use the reciprocal power

eq=[(x + 1)^\frac{2}{3}]^\frac{3}{2} = [4]^\frac{3}{2} <-- take the square root of 4 and cube it

 

eq=x + 1 = \pm8

 

eq=x = -1 \pm8

 

eq=x = -9, 7

 

 

 

Ex. eq=\sqrt{2x+6}-\sqrt{x+4}=1

 

eq=\sqrt{2x+6}=1+\sqrt{x+4} <--1 radical on each side, now square each side (you have to foil the right side)

 

eq=2x+6=1+2\sqrt{x+4}+x+4 <-- combine like terms (4 and 1)

 

eq=2x+6=5+x+2\sqrt{x+4} <-- isolate the radical

 

eq=x+1=2\sqrt{x+4} <-- square both sides to get rid of the radical

 

eq=x^2+2x+1=4x+16 <-- get everything on one side

 

eq=x^2-2x-15=0 <-- factor it

 

eq=(x-5)(x+3)

 

eq=x=5, -3

 

 

 

 

Ex. eq=\sqrt{2x+7}-x=2

 

eq=\sqrt{2x+7}=x+2

 

eq=2x+7=x^2+4x+4

 

eq=0=x^2+2x-3

 

eq=(x+3)(x-1)

 

eq=x=-3,1  but if you plug in -3, it doesn't work, so...

 

eq=x=1

 

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Comments (1)

Anonymous said

at 8:15 pm on Jan 8, 2008

Nice job, Megan and Laura! 20/20

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