+0

# Hey so I have another question

0
196
2
+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 $a + b + c$?

WhichWitchIsWhich  Oct 22, 2017

#1
+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
+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
#2
+272
+2

thanks so much that was super helpful

WhichWitchIsWhich  Oct 22, 2017