Question 197874
# 1




{{{t^2-2t=15}}} Start with the given equation.



{{{t^2-2t-15=0}}} Get all terms to the left side.



Notice we have a quadratic in the form of {{{At^2+Bt+C}}} where {{{A=1}}}, {{{B=-2}}}, and {{{C=-15}}}



Let's use the quadratic formula to solve for "t":



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



{{{t = (-(-2) +- sqrt( (-2)^2-4(1)(-15) ))/(2(1))}}} Plug in  {{{A=1}}}, {{{B=-2}}}, and {{{C=-15}}}



{{{t = (2 +- sqrt( (-2)^2-4(1)(-15) ))/(2(1))}}} Negate {{{-2}}} to get {{{2}}}. 



{{{t = (2 +- sqrt( 4-4(1)(-15) ))/(2(1))}}} Square {{{-2}}} to get {{{4}}}. 



{{{t = (2 +- sqrt( 4--60 ))/(2(1))}}} Multiply {{{4(1)(-15)}}} to get {{{-60}}}



{{{t = (2 +- sqrt( 4+60 ))/(2(1))}}} Rewrite {{{sqrt(4--60)}}} as {{{sqrt(4+60)}}}



{{{t = (2 +- sqrt( 64 ))/(2(1))}}} Add {{{4}}} to {{{60}}} to get {{{64}}}



{{{t = (2 +- sqrt( 64 ))/(2)}}} Multiply {{{2}}} and {{{1}}} to get {{{2}}}. 



{{{t = (2 +- 8)/(2)}}} Take the square root of {{{64}}} to get {{{8}}}. 



{{{t = (2 + 8)/(2)}}} or {{{t = (2 - 8)/(2)}}} Break up the expression. 



{{{t = (10)/(2)}}} or {{{t =  (-6)/(2)}}} Combine like terms. 



{{{t = 5}}} or {{{t = -3}}} Simplify. 



So the solutions are {{{t = 5}}} or {{{t = -3}}} 

  
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# 2


{{{0.9a^2+5.4a-4=0}}} Start with the given equation.



{{{9a^2+54a-40=0}}} Multiply EVERY term by 10 to make every number a whole number.



Notice we have a quadratic in the form of {{{Aa^2+Ba+C}}} where {{{A=9}}}, {{{B=54}}}, and {{{C=-40}}}



Let's use the quadratic formula to solve for "a":



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



{{{a = (-(54) +- sqrt( (54)^2-4(9)(-40) ))/(2(9))}}} Plug in  {{{A=9}}}, {{{B=54}}}, and {{{C=-40}}}



{{{a = (-54 +- sqrt( 2916-4(9)(-40) ))/(2(9))}}} Square {{{54}}} to get {{{2916}}}. 



{{{a = (-54 +- sqrt( 2916--1440 ))/(2(9))}}} Multiply {{{4(9)(-40)}}} to get {{{-1440}}}



{{{a = (-54 +- sqrt( 2916+1440 ))/(2(9))}}} Rewrite {{{sqrt(2916--1440)}}} as {{{sqrt(2916+1440)}}}



{{{a = (-54 +- sqrt( 4356 ))/(2(9))}}} Add {{{2916}}} to {{{1440}}} to get {{{4356}}}



{{{a = (-54 +- sqrt( 4356 ))/(18)}}} Multiply {{{2}}} and {{{9}}} to get {{{18}}}. 



{{{a = (-54 +- 66)/(18)}}} Take the square root of {{{4356}}} to get {{{66}}}. 



{{{a = (-54 + 66)/(18)}}} or {{{a = (-54 - 66)/(18)}}} Break up the expression. 



{{{a = (12)/(18)}}} or {{{a =  (-120)/(18)}}} Combine like terms. 



{{{a = 2/3}}} or {{{a = -20/3}}} Simplify. 



So the solutions are {{{a = 2/3}}} or {{{a = -20/3}}} 


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# 3


Are you looking for the value of "c" that will make {{{x^2+10x+c}}} a perfect square??? There are no other instructions?


If so, then take half of 10 to get 5. Square 5 to get 25. This means that {{{c=25}}}



So the expression is {{{x^2+10x+25}}} which is a perfect square.