Question 196896
# 1


{{{x^2-3x=-10}}} Start with the given equation.



{{{x^2-3x+10=0}}} Get all terms to the left side.



Notice we have a quadratic in the form of {{{Ax^2+Bx+C}}} where {{{A=1}}}, {{{B=-3}}}, and {{{C=10}}}



Let's use the quadratic formula to solve for x



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(-3) +- sqrt( (-3)^2-4(1)(10) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=-3}}}, and {{{C=10}}}



{{{x = (3 +- sqrt( (-3)^2-4(1)(10) ))/(2(1))}}} Negate {{{-3}}} to get {{{3}}}. 



{{{x = (3 +- sqrt( 9-4(1)(10) ))/(2(1))}}} Square {{{-3}}} to get {{{9}}}. 



{{{x = (3 +- sqrt( 9-40 ))/(2(1))}}} Multiply {{{4(1)(10)}}} to get {{{40}}}



{{{x = (3 +- sqrt( -31 ))/(2(1))}}} Subtract {{{40}}} from {{{9}}} to get {{{-31}}}



{{{x = (3 +- sqrt( -31 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{x = (3 +- i*sqrt(31))/(2)}}} Simplify the square root  



{{{x = (3+i*sqrt(31))/(2)}}} or {{{x = (3-i*sqrt(31))/(2)}}} Break up the expression.  



So the answers are {{{x = (3+i*sqrt(31))/(2)}}} or {{{x = (3-i*sqrt(31))/(2)}}} 


<hr>


# 2


{{{(x+8)(x-8)(x+12)>0}}}  Start with the given inequality



{{{(x+8)(x-8)(x+12)=0}}} Set the left side equal to zero



Set each individual factor equal to zero:


{{{x+8=0}}}, {{{x-8=0}}} or {{{x+12=0}}}


Solve for x in each case:


{{{x=-8}}}, {{{x=8}}} or {{{x=-12}}}



So our critical values are {{{x=-8}}}, {{{x=8}}} and {{{x=-12}}}


Now set up a number line and plot the critical values on the number line


{{{number_line( 600, -15, 10,-8,8,-12)}}}




So let's pick some test points that are near the critical values and evaluate them.



Let's pick a test value that is less than {{{-12}}} (notice how it's to the left of the leftmost endpoint):


So let's pick {{{x=-13}}}


{{{(x+8)(x-8)(x+12)>0}}} Start with the given inequality



{{{(-13+8)(-13-8)(-13+12)> 0}}} Plug in {{{x=-13}}}



{{{-105> 0}}} Evaluate and simplify the left side


Since the inequality is false, this means that the interval does <b>not</b> work. So this interval is <b>not</b> in our solution set and we can ignore it.



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Let's pick a test value that is in between {{{-12}}} and {{{-8}}}:


So let's pick {{{x=-10}}}


{{{(x+8)(x-8)(x+12)>0}}} Start with the given inequality



{{{(-10+8)(-10-8)(-10+12)> 0}}} Plug in {{{x=-10}}}



{{{72> 0}}} Evaluate and simplify the left side


Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.

   So part our solution in interval notation is <font size="8">(</font>*[Tex \LARGE -12,-8]<font size="8">)</font>



   



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Let's pick a test value that is in between {{{-8}}} and {{{8}}}:


So let's pick {{{x=0}}}


{{{(x+8)(x-8)(x+12)>0}}} Start with the given inequality



{{{(0+8)(0-8)(0+12)> 0}}} Plug in {{{x=0}}}



{{{-768> 0}}} Evaluate and simplify the left side


Since the inequality is false, this means that the interval does <b>not</b> work. So this interval is <b>not</b> in our solution set and we can ignore it.



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Let's pick a test value that is greater than {{{8}}} (notice how it's to the right of the rightmost endpoint):


So let's pick {{{x=9}}}


{{{(x+8)(x-8)(x+12)>0}}} Start with the given inequality



{{{(9+8)(9-8)(9+12)> 0}}} Plug in {{{x=9}}}



{{{357> 0}}} Evaluate and simplify the left side


Since the inequality is true, this means that the interval works. So this tells us that this interval is in our solution set.

   So part our solution in interval notation is <font size="8">(</font>*[Tex \LARGE 8,\infty]<font size="8">)</font>



   



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Summary:


So the solution in interval notation is:



<font size="8">(</font>*[Tex \LARGE -12,-8]<font size="8">)</font> *[Tex \LARGE \cup] <font size="8">(</font>*[Tex \LARGE 8,\infty]<font size="8">)</font>




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# 3


{{{2x^2-5x+1}}} Start with the given expression.



{{{2(-4)^2-5(-4)+1}}} Plug in {{{x=-4}}}.



{{{2(16)-5(-4)+1}}} Square {{{-4}}} to get {{{16}}}.



{{{32-5(-4)+1}}} Multiply {{{2}}} and {{{16}}} to get {{{32}}}.



{{{32+20+1}}} Multiply {{{-5}}} and {{{-4}}} to get {{{20}}}.



{{{53}}} Combine like terms.



So {{{2x^2-5x+1=53}}} when {{{x=-4}}}


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# 4





{{{(9x^8-3)(x^9-3)}}} Start with the given expression.



Now let's FOIL the expression.



Remember, when you FOIL an expression, you follow this procedure:



{{{(highlight(9x^8)-3)(highlight(x^9)-3)}}} Multiply the <font color="red">F</font>irst terms:{{{(9*x^8)*(x^9)=9*x^17}}}. Note: simply add the exponents



{{{(highlight(9x^8)-3)(x^9+highlight(-3))}}} Multiply the <font color="red">O</font>uter terms:{{{(9*x^8)*(-3)=-27*x^8}}}.



{{{(9x^8+highlight(-3))(highlight(x^9)-3)}}} Multiply the <font color="red">I</font>nner terms:{{{(-3)*(x^9)=-3*x^9}}}.



{{{(9x^8+highlight(-3))(x^9+highlight(-3))}}} Multiply the <font color="red">L</font>ast terms:{{{(-3)*(-3)=9}}}.



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

So we have the terms: {{{9*x^17}}}, {{{-27*x^8}}}, {{{-3*x^9}}}, {{{9}}} 



{{{9*x^17-27*x^8-3*x^9+9}}} Now add every term listed above to make a single expression.



So {{{(9x^8-3)(x^9-3)}}} FOILs to {{{9*x^17-27*x^8-3*x^9+9}}}.



In other words, {{{(9x^8-3)(x^9-3)=9*x^17-27*x^8-3*x^9+9}}}.