Total math brain farts, help? Especially math teachers?

Question

I'm in pre-calc now and today was the first class and we got a algebra 2 review sheet for homework. I got most of them, just three of them I forgot about. I will take answers with work but quick refreshers on why you did certain things could be helpful. (not full explainations for each step, I'll catch on)

1. Solve the equation: 3x^2-6x=1

2. What is the equation of the line passing through the point (0,2) and perpendicular to the line y=2/3x+8

3. Find the quotient when 4x^3+12x^2-14x+8 is divided by x+4

Thanks!

Answer 1

1)

Put the equation into a standard quadratic form by subtracting 1 from each side.

3x^2 - 6x -1 = 0

Now, use the quadratic formula ( (-b +-  sqrt(b^2 - 4ac))/2a)

(6 +- sqrt(48))/6

(6 +- 4*sqrt(3))/6

x = 1 +2sqrt(3)/3

or

x = 1 - 2sqrt(3)/3

-----------------------------

2) The slope of a line perpendicular to another line is equal to its negative reciprocal.

given y = 2/3x + 8

the perpendicular slope is -3/2

so

y = -3/2x + b

Plug in the given [pint

0 = -3/2*2 + b

0 = -3 + b

3 = b

So the equation is...

y = -3/2x + 3

--------------------------------------...

3)

You need to do polynomial long division

...........4x^2 - 4x + 2

.........________________

x + 4 ) 4x^3 + 12x^2 - 14x + 8

............4x^3 + 16x^2

............------------------

........................-4x^2 - 14x

........................-4x^2 - 16x

........................--------------...

.....................................2... + 8

.....................................2... + 8

.....................................-...

.........................................

So,

4x^3 + 12x^2 - 14x + 8

divided by

x + 4

is

4x^2 - 4x + 2

Answer 2

1.  This is 3x^2 - 6x - 1 = 0.  Use the quadratic formula.

2. Remember that perpendicular lines have slopes that are negative reciprocals of each other.  So you want the line with a slope of -3/2 that includes (0,2).  You're given the slope and the y-intercept, so substitute those in for m and b into y = mx + b.

3. Go here to learn polynomial long division:

http://mathforum.org/library/drmath/view...

Answer 3

1) This is a quadratic equation so bring the 1 to the other side so you get

3x^2-6x-1=0  

Now use the calculator to find the values of X

2) The line is perpendicular so the gradient will be inverse...no the gradient of the perpendicular will be - 3/2

so then u use formula y-y1=m(x-x1)

so y-2= -3/2(x-0)

2y-4= -3x

Equation of the prependicular passing thought (0,2) will be Y= -3/2X +2

and i dont know how to do the third one :D..lol

Answer 4

1,) for this sort of thing, rewrite the equation such that is is 'equal to zero'

3x^2 -6x -1 = 0

then see if you can 'factor it easily into'

(x + a) * (x + b) = 0

because then the 'solutions will be -a and -b  (since x either to either of those will result in the product of 'something' by zero, which will be zero.

if you can't factor it (and that's an art, aided by your memorizing lots of patterns), use the 'quadratic equation' to solve it.

To expand on factoring, what we really have is more like

(ax + b) * (cx + d ) =  0

we can multiply that out to get

acx^2 + (bc+ad)x + bd = 0

so we're trying to find values for a, b, c, d where

ac=3

bc+ad = -6

bd = -1

(coefficients from your original equation)

because of that 3, we know a and c are not both 1, so we start with something like

(3x + b) (x + d)

which is just optimism.. but now we look for values of b and d which when multiplied together are -1, but when added (with the 3/1 weighting) are -6

honestly, that doesn't look too promising.  At this point I would just run off to the quadratic equation

--------------

2.) two dimensional lines are perpendicular when they are 'at right angles' to each other, which ultimately resolves to their having inverse slopes.  so a line of slope m, is perpendicular to a line of slope -m  (has exactly the opposite change in y for a given change in x)

that turns this into "what is the equation of a line with a slope of -2/3, and which passes through (0,2)"  which I hope is easier for you.

----

3.) I don't recall the structured method for dividing polynomials, but I imagine you could sort of do it the same as 'long division' where instead of the hundreds, tens, and ones, place you have the x-cubed, x-squared, and x place.  and x^0 (ones) place, I guess

doing that on a separate piece of paper... that actually looks like a workable strategy and I get

4x^2 - 4x + 2

A bit of a head-spinner, but algebra is all about symbolic replacement of one symbol with another, whether that is a letter for a number, or some combination of letters and numbers for another combination of letters and numbers.

Of course I do not claim to have gotten that right.  Just a suggestion, your mileage may vary

--

clearly I type too slowly :-)

Answer 5

bbooooooooooooooooo

Answer 6

1) Just use the formula to solve a quadratic equation ax^2 + bx + c=0

The formula is  x = (-b +- sqrt(b^2 -4ac))/2a

3x^2 -6x - 1 =0 ==>   x = (6 +- sqrt(36 +12))/6  ==>

x = (6 + sqrt(48))/6  or  x = (6- sqrt(48))/6  you can simplify.

2) To be perpendicular de slope must be the inverse and simetric

   As y = (2/3)x + 8  the slope of the perpendicular is  -3/2

The equation is  y = (-3/2)x + b  To find b you put the point in the equation. That is when x=0, y =2 ==>   2 = (-3/2)0 + b ==> b=2

The equation of the perpendicular is  y = (-3/2)x + 2

3) Use Briot-Ruffini rule

     4   12   -14   8

-4  4   -4     2     0    you must study it in a book

The answer  is  4x^2 -4x + 2  

Answer 7

1)  3x2 - 6x = 1

3x^2 - 6x - 1 = 0

A quadratic equation with a = 3, b = -6, and c = -1

x = [-b +/- sqrt(b^2 - 4ac)] / (2a)

x = [6 +/- sqrt(36 - 4(3)(-1)] / (2*3)

x = (6 +/- sqrt(48)) / 6

x = (6 +/- 4sqrt(3)) / 6

x = 1 +/- 2sqrt(3) / 3

2)  Perpendicular to y = 2/3 x + 8 means the slope will be -3/2

passing through (0,2) means the y-intercept is 2

the equation will be y = (-3/2) x + 2

3)  either by synthetic or polynomial long division:

-4 |...4.....12 .....-14 .....8

...............-16... ...16....-8

--------------------------------

........4... ..-4..........2.......0

so 4x^3 + 12x^2 - 14x + 8 = (x + 4)(4x^2 - 4x + 2)

quotient is 4x^2 - 4x + 2

(4x^3 + 12x^2 - 14x + 8) / (x + 4) = 4x^2 - 4x + 2