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avatar+272 

Hey so I have another question

 

Patricia is trying to solve the following equation by completing the square: $$25x^2+20x-10 = 0.$$She successfully rewrites the above equation in the following form: $$(ax + b)^2 = c,$$where $a,$ $b,$ and $c$ are integers and $a > 0.$ What is the value of $a + b + c$?

 

Thanks for your help

WhichWitchIsWhich  Oct 22, 2017

Best Answer 

 #1
avatar+7068 
+1

25x2 + 20x - 10  =  0      First, divide through by  25 .

 

\(\frac{25}{25}\)x2  +  \(\frac{20}{25}\)x - \(\frac{10}{25}\)  =  \(\frac{0}{25}\)

 

x2  +  \(\frac45\)x - \(\frac25\)  =  0          Add  \(\frac25\)  to both sides of the equation.

 

x2  +  \(\frac45\)x  =  \(\frac25\)               Now add  (\(\frac4{10}\))2  to both sides of the equation.

 

x2  +  \(\frac45\)x + (\(\frac4{10}\))2  =  \(\frac25\) + (\(\frac4{10}\))2       Factor the left side and simplify the right side.

 

(x + \(\frac4{10}\))2  =  \(\frac{40}{100}\)+\(\frac{16}{100}\)

 

(x + \(\frac4{10}\))2  =  \(\frac{56}{100}\)

 

Now, it said a, b, and c have to be integers...let's multiply both sides by  102

 

102 * (x + \(\frac4{10}\))2  =  102 * \(\frac{56}{100}\)         And   a2b2  =  (ab)2 .

 

( 10 * (x + \(\frac4{10}\)) )2  =  100 * \(\frac{56}{100}\)       Distribute the  10 .

 

(10x + 4)2  =  56

 

So....  a + b + c  =  10 + 4 + 56  =  70

hectictar  Oct 22, 2017
 #1
avatar+7068 
+1
Best Answer

25x2 + 20x - 10  =  0      First, divide through by  25 .

 

\(\frac{25}{25}\)x2  +  \(\frac{20}{25}\)x - \(\frac{10}{25}\)  =  \(\frac{0}{25}\)

 

x2  +  \(\frac45\)x - \(\frac25\)  =  0          Add  \(\frac25\)  to both sides of the equation.

 

x2  +  \(\frac45\)x  =  \(\frac25\)               Now add  (\(\frac4{10}\))2  to both sides of the equation.

 

x2  +  \(\frac45\)x + (\(\frac4{10}\))2  =  \(\frac25\) + (\(\frac4{10}\))2       Factor the left side and simplify the right side.

 

(x + \(\frac4{10}\))2  =  \(\frac{40}{100}\)+\(\frac{16}{100}\)

 

(x + \(\frac4{10}\))2  =  \(\frac{56}{100}\)

 

Now, it said a, b, and c have to be integers...let's multiply both sides by  102

 

102 * (x + \(\frac4{10}\))2  =  102 * \(\frac{56}{100}\)         And   a2b2  =  (ab)2 .

 

( 10 * (x + \(\frac4{10}\)) )2  =  100 * \(\frac{56}{100}\)       Distribute the  10 .

 

(10x + 4)2  =  56

 

So....  a + b + c  =  10 + 4 + 56  =  70

hectictar  Oct 22, 2017
 #2
avatar+272 
+2

thanks so much that was super helpful

WhichWitchIsWhich  Oct 22, 2017

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