Question 170193
# 1



{{{7v/(v^2-49) + v/(v-7)}}} Start with the given equation



{{{7v/((v-7)(v+7)) + v/(v-7)}}} Factor the first denominator



Take note that the LCD is {{{(v-7)(v+7)}}}



{{{7v/((v-7)(v+7)) + (v(v+7))/((v-7)(v+7))}}} Multiply the second fraction by {{{(v+7)/(v+7)}}} to make the denominators equal.



{{{7v/((v-7)(v+7)) + (v^2+7v)/((v-7)(v+7))}}} Distribute



{{{(7v+v^2+7v)/((v-7)(v+7))}}} Combine the fractions.



{{{(v^2+14v)/((v-7)(v+7))}}} Combine like terms.



{{{(v^2+14v)/(v^2-49)}}} FOIL the denominator.



So {{{7v/(v^2-49) + v/(v-7)}}} simplifies to {{{(v^2+14v)/(v^2-49)}}} 



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# 2



Area: {{{A=Length*Width=z(z+8)=z^2+8z}}}



So the area is {{{A=z^2+8z}}} square units



Perimeter: {{{P=2*length+2*width=2(z)+2(z+8)=2z+2z+16=4z+16}}}



So the perimeter is {{{P=4z+16}}} units



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# 3



{{{sqrt(80)-6*sqrt(5)}}} Start with the given expression



{{{4*sqrt(5)-6*sqrt(5)}}} Simplify {{{sqrt(80)}}} to get {{{4*sqrt(5)}}}. Note: If you need help with simplifying square roots, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>.



Since we have the common term {{{sqrt(5)}}}, we can combine like terms



{{{(4-6)sqrt(5)}}} Combine like terms. Remember, {{{5x+3x-4x=(5+3-4)x=4x}}}



{{{-2*sqrt(5)}}} Now simplify {{{4-6}}} to get {{{-2}}}



So {{{sqrt(80)-6*sqrt(5)}}} simplifies to {{{-2*sqrt(5)}}}. 



In other words,  {{{sqrt(80)-6*sqrt(5)=-2*sqrt(5)}}}



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# 4



Area of original square {{{A[1]=s^2}}}



Area of new square: {{{A[2]=(s+7)^2}}}



Since the "area becomes 196cm^2", this means that {{{A[2]=196}}}



{{{A[2]=(s+7)^2}}} Start with the second equation



{{{196=(s+7)^2}}} Plug in {{{A[2]=196}}}



{{{196=s^2+14s+49}}} FOIL



{{{0=s^2+14s+49-196}}} Subtract 196 from both sides.



{{{0=s^2+14s-147}}} Combine like terms.



Notice we have a quadratic equation in the form of {{{as^2+bs+c}}} where {{{a=1}}}, {{{b=14}}}, and {{{c=-147}}}



Let's use the quadratic formula to solve for s



{{{s = (-b +- sqrt( b^2-4ac ))/(2a)}}} Start with the quadratic formula



{{{s = (-(14) +- sqrt( (14)^2-4(1)(-147) ))/(2(1))}}} Plug in  {{{a=1}}}, {{{b=14}}}, and {{{c=-147}}}



{{{s = (-14 +- sqrt( 196-4(1)(-147) ))/(2(1))}}} Square {{{14}}} to get {{{196}}}. 



{{{s = (-14 +- sqrt( 196--588 ))/(2(1))}}} Multiply {{{4(1)(-147)}}} to get {{{-588}}}



{{{s = (-14 +- sqrt( 196+588 ))/(2(1))}}} Rewrite {{{sqrt(196--588)}}} as {{{sqrt(196+588)}}}



{{{s = (-14 +- sqrt( 784 ))/(2(1))}}} Add {{{196}}} to {{{588}}} to get {{{784}}}



{{{s = (-14 +- sqrt( 784 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{s = (-14 +- 28)/(2)}}} Take the square root of {{{784}}} to get {{{28}}}. 



{{{s = (-14 + 28)/(2)}}} or {{{s = (-14 - 28)/(2)}}} Break up the expression. 



{{{s = (14)/(2)}}} or {{{s =  (-42)/(2)}}} Combine like terms. 



{{{s = 7}}} or {{{s = -21}}} Simplify. 



So the possible answers are {{{s = 7}}} or {{{s = -21}}} 



However, since a negative side length is NOT possible, this means that the only answer is {{{s = 7}}}



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Answer:


So the original side length is 7 centimeters