Question 110812
#1




*[Tex \LARGE (x+4)^2=7] Start with the given equation




*[Tex \LARGE x+4=\pm sqrt{7}] Take the square root of both sides




*[Tex \LARGE x=-4\pm sqrt{7}] Subtract 4 from both sides to isolate x.




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

Answer:


So our solution is:

*[Tex \LARGE x=-4\pm sqrt{7}]



This means our solution breaks down to:


*[Tex \LARGE x=-4+sqrt{7}]


or


*[Tex \LARGE x=-4-sqrt{7}]


<hr>


#2




{{{x^2+7x}}} Start with the given equation


Take half of the x coefficient 7 to get 3.5 (ie {{{7/2=3.5}}})

Now square 3.5 to get 12.25 (ie {{{(3.5)^2=12.25}}})




{{{x^2+7x+12.25}}} Add this result (12.25) to the expression. Now the expression {{{x^2+7x+12.25}}} is a perfect square trinomial.


<hr>


#3



In order to graph {{{y=-x^2+6x}}}, we need to plot some points.


We can start at any x value, so lets start at x=1





Lets evaluate {{{f(1)}}}


{{{f(x)=-x^2+6x}}} Start with the given polynomial



{{{f(1)=-(1)^2+6(1)}}} Plug in {{{x=1}}}



{{{f(1)=-1+6(1)}}} Evaluate {{{(1)^2}}} to get 1

 

{{{f(1)=-1+6}}} Multiply 6 and  1  to get 6

  

{{{f(1)=5}}} Now combine like terms



So when {{{x=1}}}, {{{f(x)=5}}}




-------Now lets find another point-------




Lets evaluate {{{f(2)}}}


{{{f(x)=-x^2+6x}}} Start with the given polynomial



{{{f(2)=-(2)^2+6(2)}}} Plug in {{{x=2}}}



{{{f(2)=-4+6(2)}}} Evaluate {{{-1(2)^2}}} to get -14

 

{{{f(2)=-4+12}}} Multiply 6 and  2  to get 12

  

{{{f(2)=8}}} Now combine like terms



So when {{{x=2}}}, {{{f(x)=8}}}




-------Now lets find another point-------




Lets evaluate {{{f(3)}}}


{{{f(x)=-x^2+6x}}} Start with the given polynomial



{{{f(3)=-(3)^2+6(3)}}} Plug in {{{x=3}}}



{{{f(3)=-9+6(3)}}} Evaluate {{{-1(3)^2}}} to get -19

 

{{{f(3)=-9+18}}} Multiply 6 and  3  to get 18

  

{{{f(3)=9}}} Now combine like terms



So when {{{x=3}}}, {{{f(x)=9}}}




-------Now lets find another point-------




Lets evaluate {{{f(4)}}}


{{{f(x)=-x^2+6x}}} Start with the given polynomial



{{{f(4)=-(4)^2+6(4)}}} Plug in {{{x=4}}}



{{{f(4)=-16+6(4)}}} Evaluate {{{-1(4)^2}}} to get -116

 

{{{f(4)=-16+24}}} Multiply 6 and  4  to get 24

  

{{{f(4)=8}}} Now combine like terms



So when {{{x=4}}}, {{{f(x)=8}}}




-------Now lets find another point-------




Lets evaluate {{{f(5)}}}


{{{f(x)=-x^2+6x}}} Start with the given polynomial



{{{f(5)=-(5)^2+6(5)}}} Plug in {{{x=5}}}



{{{f(5)=-25+6(5)}}} Evaluate {{{-1(5)^2}}} to get -125

 

{{{f(5)=-25+30}}} Multiply 6 and  5  to get 30

  

{{{f(5)=5}}} Now combine like terms



So when {{{x=5}}}, {{{f(x)=5}}}



Now lets make a table of the values we have calculated

<pre>
<TABLE width=500>

<TR><TD> x</TD><TD>y</TD></TR>

<TR><TD> 1</TD><TD>5</TD></TR> 
<TR><TD> 2</TD><TD>8</TD></TR> 
<TR><TD> 3</TD><TD>9</TD></TR> 
<TR><TD> 4</TD><TD>8</TD></TR> 
<TR><TD> 5</TD><TD>5</TD></TR> 
</TABLE>
</pre>Now plot the points

{{{drawing(900,900,-10,10,-10,10,
  grid( 1 ),
circle(1,5,0.05),
circle(1,5,0.08),
circle(2,8,0.05),
circle(2,8,0.08),
circle(3,9,0.05),
circle(3,9,0.08),
circle(4,8,0.05),
circle(4,8,0.08),
circle(5,5,0.05),
circle(5,5,0.08))}}}



Now connect the points to graph {{{y=-x^2+6x}}}  (note: the more points you plot, the easier it is to draw the graph)

{{{drawing(900,900,-10,10,-10,10,
grid( 1 ),
graph(900,900,-10,10,-10,10, -x^2+6x),
circle(1,5,0.05),
circle(1,5,0.08),
circle(2,8,0.05),
circle(2,8,0.08),
circle(3,9,0.05),
circle(3,9,0.08),
circle(4,8,0.05),
circle(4,8,0.08),
circle(5,5,0.05),
circle(5,5,0.08))}}}