Po is trying to solve the following equation by completing the square:$$49x^2+56x-64 = 0.$$He 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 $a + b + c$?

Question

Mathematics

Rory Bowman

17 February, 04:39

Po is trying to solve the following equation by completing the square:$$49x^2+56x-64 = 0.$$He 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 $a + b + c$?

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Silvester · Answer 1

The question is ...

Po is trying to solve the following equation by completing the square: 49x^2+56x-64 = 0. He 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?

Answer:

3.122

Step-by-step explanation:

49x^2+56x-64 = 0

49 (x^2+56x/49) - 64 = 0

49 (x^2+8x/7) = 60

introduce a new constant to form (ax+b) ^2 form from the multiplier of x

New constant = ((8/7) : 2) ^2

= (8/14) ^2

Put the constant into equation plus another equal constant with negative value

49 (x^2 + 8x/7 + (8/14) ^2 - (8/14) ^2) = 60

Reduce the following part

x^2 + 8x/7 + (8/14) ^2 = (x + 8/14) ^2

Put back into equation:

49 ((x + 8/14) ^2) - 49 (8/14) ^2 = 60

49 ((x + 8/14) ^2) = 60 + 49 (8/14) ^2

= 60 + 49 (64/196)

= 60 + 16 = 76

(x + 8/14) ^2 = 76/49

Equating with (ax + b) ^2 = c

a = 1, b = 8/14, c = 76/49

a + b + c = 1 + 8/14 + 76/49 = 3.122