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Let x$ be a positive number such that 2x^2 = 4x + 9.$ If $x$ can be written in simplified form as $\dfrac{a + \sqrt{b}}{c}$ such that $a,$ $b,$ and $c$ are positive integers, what is $a + b + c$?

 May 31, 2019
 #1
avatar+25 
-2

uhhh can you put it in latex????????????? wait lemme do that: 

 

Let x be a positive number such that 2x^2 = 4x + 9. If x can be written in simplified form as\( \dfrac{a + \sqrt{b}}{c} \) such that a, b, and c are positive integers, what is a + b + c?

 May 31, 2019
 #2
avatar+25 
-2

ummm so i got 

 

a=4

b=88

c=4

so if you add ummm

aww i cant add in my head..... uhhhhhhhhh 88+8=96?

i got 96. it should be right.....

ProffesorNobody  May 31, 2019
 #3
avatar+25 
-3

96 is the answerrrrr 

ProffesorNobody  Jun 1, 2019
 #6
avatar+8829 
+3

\(\sqrt{88}\)   can be simplified to   \(2\sqrt{22}\)   because   \(\sqrt{88}\ =\ \sqrt{4\cdot22}\ =\ \sqrt4\cdot\sqrt{22}\ =\ 2\sqrt{22}\)

 

 

But it is true that    x = \(\frac{4+\sqrt{88}}{4}\)     smiley

hectictar  Jun 1, 2019
 #8
avatar+25 
-1

Thank you. I understnad why i was wrong...

ProffesorNobody  Jun 1, 2019
 #4
avatar
+1

nope

 Jun 1, 2019
 #5
avatar+8829 
+3

2x2  =  4x + 9

                               Subtract  4x  from both sides of the equation.

2x2 - 4x  =  9

                               Subtract  9  from both sides of the equation.

2x2 - 4x - 9  =  0

                               By the quadratic formula,

 

x  =  \({-(-4) \pm \sqrt{(-4)^2-4(2)(-9)} \over 2(2)}\ =\ \frac{4\pm\sqrt{16+72}}{4}\ =\ \frac{4\pm\sqrt{88}}{4}\ =\ \frac{4\pm2\sqrt{22}}{4}\ =\ \frac{2\pm\sqrt{22}}{2}\)

 

We're given that  x  is positive so....

 

x  =  \(\frac{2+\sqrt{22}}{2}\)

 

Now it is in simplified form as   \(\frac{a+\sqrt{b}}{c}\)   where  a,  b,  and  c  are positive integers.

 

a + b + c   =   2 + 22 + 2   =   26

 Jun 1, 2019
 #7
avatar+25 
-1

i used the quadratic formula..... what did i do wrong

ProffesorNobody  Jun 1, 2019
edited by ProffesorNobody  Jun 1, 2019
 #9
avatar+7747 
0

\(\begin{array}{rcll} 2x^2 &=& 4x + 9\\ 2x^2 - 4x - 9 &=& 0\\ x &=&\dfrac{4\pm\sqrt{16+4\cdot2\cdot9}}{4}\\ x &=& \dfrac{4\pm2\sqrt{22}}4\\ x &=& \dfrac{2\pm\sqrt{22}}2\\ a + b + c &=&2 + 22 + 2 \\ a + b + c &=& 26 \end{array}\)

.
 Jun 1, 2019

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