```Question 147136
# 1: Solve the equation 3x2 - 15x = 0.

Let's use the quadratic formula to solve for x

{{{x = (-(-15) +- sqrt( (-15)^2-4(3)(0) ))/(2(3))}}} Plug in  {{{a=3}}}, {{{b=-15}}}, and {{{c=0}}}

{{{x = (15 +- sqrt( (-15)^2-4(3)(0) ))/(2(3))}}} Negate {{{-15}}} to get {{{15}}}.

{{{x = (15 +- sqrt( 225-4(3)(0) ))/(2(3))}}} Square {{{-15}}} to get {{{225}}}.

{{{x = (15 +- sqrt( 225-0 ))/(2(3))}}} Multiply {{{4(3)(0)}}} to get {{{0}}}

{{{x = (15 +- sqrt( 225 ))/(2(3))}}} Subtract {{{0}}} from {{{225}}} to get {{{225}}}

{{{x = (15 +- sqrt( 225 ))/(6)}}} Multiply {{{2}}} and {{{3}}} to get {{{6}}}.

{{{x = (15 +- 15)/(6)}}} Take the square root of {{{225}}} to get {{{15}}}.

{{{x = (15 + 15)/(6)}}} or {{{x = (15 - 15)/(6)}}} Break up the expression.

{{{x = (30)/(6)}}} or {{{x =  (0)/(6)}}} Combine like terms.

{{{x = 5}}} or {{{x = 0}}} Simplify.

So our answers are {{{x = 5}}} or {{{x = 0}}}

<hr>

# 2: Solve the equation 6x2 - 5x = 6

{{{6x^2-5x-6=0}}} Get all terms to the left side.

Notice we have a quadratic equation in the form of {{{ax^2+bx+c}}} where {{{a=6}}}, {{{b=-5}}}, and {{{c=-6}}}

Let's use the quadratic formula to solve for x

{{{x = (-(-5) +- sqrt( (-5)^2-4(6)(-6) ))/(2(6))}}} Plug in  {{{a=6}}}, {{{b=-5}}}, and {{{c=-6}}}

{{{x = (5 +- sqrt( (-5)^2-4(6)(-6) ))/(2(6))}}} Negate {{{-5}}} to get {{{5}}}.

{{{x = (5 +- sqrt( 25-4(6)(-6) ))/(2(6))}}} Square {{{-5}}} to get {{{25}}}.

{{{x = (5 +- sqrt( 25--144 ))/(2(6))}}} Multiply {{{4(6)(-6)}}} to get {{{-144}}}

{{{x = (5 +- sqrt( 25+144 ))/(2(6))}}} Rewrite {{{sqrt(25--144)}}} as {{{sqrt(25+144)}}}

{{{x = (5 +- sqrt( 169 ))/(2(6))}}} Add {{{25}}} to {{{144}}} to get {{{169}}}

{{{x = (5 +- sqrt( 169 ))/(12)}}} Multiply {{{2}}} and {{{6}}} to get {{{12}}}.

{{{x = (5 +- 13)/(12)}}} Take the square root of {{{169}}} to get {{{13}}}.

{{{x = (5 + 13)/(12)}}} or {{{x = (5 - 13)/(12)}}} Break up the expression.

{{{x = (18)/(12)}}} or {{{x =  (-8)/(12)}}} Combine like terms.

{{{x = 3/2}}} or {{{x = -2/3}}} Simplify.

So our answers are {{{x = 3/2}}} or {{{x = -2/3}}}

<hr>

# 3: Factor the expression x2 - 8xy + 12y2 completely.

Looking at {{{1x^2-8xy+12y^2}}} we can see that the first term is {{{1x^2}}} and the last term is {{{12y^2}}} where the coefficients are 1 and 12 respectively.

Now multiply the first coefficient 1 and the last coefficient 12 to get 12. Now what two numbers multiply to 12 and add to the  middle coefficient -8? Let's list all of the factors of 12:

Factors of 12:

1,2,3,4,6,12

-1,-2,-3,-4,-6,-12 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 12

1*12

2*6

3*4

(-1)*(-12)

(-2)*(-6)

(-3)*(-4)

note: remember two negative numbers multiplied together make a positive number

Now which of these pairs add to -8? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -8

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">12</td><td>1+12=13</td></tr><tr><td align="center">2</td><td align="center">6</td><td>2+6=8</td></tr><tr><td align="center">3</td><td align="center">4</td><td>3+4=7</td></tr><tr><td align="center">-1</td><td align="center">-12</td><td>-1+(-12)=-13</td></tr><tr><td align="center">-2</td><td align="center">-6</td><td>-2+(-6)=-8</td></tr><tr><td align="center">-3</td><td align="center">-4</td><td>-3+(-4)=-7</td></tr></table>

From this list we can see that -2 and -6 add up to -8 and multiply to 12

Now looking at the expression {{{1x^2-8xy+12y^2}}}, replace {{{-8xy}}} with {{{-2xy+-6xy}}} (notice {{{-2xy+-6xy}}} adds up to {{{-8xy}}}. So it is equivalent to {{{-8xy}}})

{{{1x^2+highlight(-2xy+-6xy)+12y^2}}}

Now let's factor {{{1x^2-2xy-6xy+12y^2}}} by grouping:

{{{(1x^2-2xy)+(-6xy+12y^2)}}} Group like terms

{{{x(x-2y)-6y(x-2y)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-6y}}} out of the second group

{{{(x-6y)(x-2y)}}} Since we have a common term of {{{x-2y}}}, we can combine like terms

So {{{1x^2-2xy-6xy+12y^2}}} factors to {{{(x-6y)(x-2y)}}}

So this also means that {{{1x^2-8xy+12y^2}}} factors to {{{(x-6y)(x-2y)}}} (since {{{1x^2-8xy+12y^2}}} is equivalent to {{{1x^2-2xy-6xy+12y^2}}})

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

So {{{x^2-8xy+12y^2}}} factors to {{{(x-6y)(x-2y)}}}

<hr>

# 4: Choose one factor of the following expression from the list below.
6x2 - 5x - 4

Looking at {{{6x^2-5x-4}}} we can see that the first term is {{{6x^2}}} and the last term is {{{-4}}} where the coefficients are 6 and -4 respectively.

Now multiply the first coefficient 6 and the last coefficient -4 to get -24. Now what two numbers multiply to -24 and add to the  middle coefficient -5? Let's list all of the factors of -24:

Factors of -24:

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to -24

(1)*(-24)

(2)*(-12)

(3)*(-8)

(4)*(-6)

(-1)*(24)

(-2)*(12)

(-3)*(8)

(-4)*(6)

note: remember, the product of a negative and a positive number is a negative number

Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5

<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">-24</td><td>1+(-24)=-23</td></tr><tr><td align="center">2</td><td align="center">-12</td><td>2+(-12)=-10</td></tr><tr><td align="center">3</td><td align="center">-8</td><td>3+(-8)=-5</td></tr><tr><td align="center">4</td><td align="center">-6</td><td>4+(-6)=-2</td></tr><tr><td align="center">-1</td><td align="center">24</td><td>-1+24=23</td></tr><tr><td align="center">-2</td><td align="center">12</td><td>-2+12=10</td></tr><tr><td align="center">-3</td><td align="center">8</td><td>-3+8=5</td></tr><tr><td align="center">-4</td><td align="center">6</td><td>-4+6=2</td></tr></table>

From this list we can see that 3 and -8 add up to -5 and multiply to -24

Now looking at the expression {{{6x^2-5x-4}}}, replace {{{-5x}}} with {{{3x+-8x}}} (notice {{{3x+-8x}}} adds up to {{{-5x}}}. So it is equivalent to {{{-5x}}})

{{{6x^2+highlight(3x+-8x)+-4}}}

Now let's factor {{{6x^2+3x-8x-4}}} by grouping:

{{{(6x^2+3x)+(-8x-4)}}} Group like terms

{{{3x(2x+1)-4(2x+1)}}} Factor out the GCF of {{{3x}}} out of the first group. Factor out the GCF of {{{-4}}} out of the second group

{{{(3x-4)(2x+1)}}} Since we have a common term of {{{2x+1}}}, we can combine like terms

So {{{6x^2+3x-8x-4}}} factors to {{{(3x-4)(2x+1)}}}

So this also means that {{{6x^2-5x-4}}} factors to {{{(3x-4)(2x+1)}}} (since {{{6x^2-5x-4}}} is equivalent to {{{6x^2+3x-8x-4}}})

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