# `sqrt(9y - 196) + sqrt196= sqrt49` Solve for y.

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The equation `sqrt(9y - 196) + sqrt 196 = sqrt 49` has to be solved for y.

`sqrt(9y - 196) + sqrt 196 = sqrt 49`

=> `sqrt(9y - 196) = sqrt 49 - sqrt 196`

`sqrt 49 = +- 7` and `sqrt 196 = +-14`

This gives:

`sqrt (9y - 196) = 21`

=> `9y - 196 = 441`

=> `y = 637/9`

`sqrt (9y - 196) = -7`

=> `9y - 196 = 49`

=> `y = 245/9`

`sqrt(9y - 196) = 7`

=> `y = 245/9`

`sqrt(9y - 196) = -21`

=> `y = 637/9`

The solutions of the equation are `245/9` and `637/9`

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You have `sqrt(9y - 196) + sqrt(196) = sqrt(49)`

Fisrt simplify first the terms that can be simplified like `sqrt(196)` and `sqrt(49)`

They are perfect square.

`sqrt(196) = 14`

`sqrt(49)=7`

`` So you have `sqrt(9y - 196) + 14 = 7`

Simplify further by combining similar terms. Move 14 to the right side, thus changing the sign to negative. Remeber that moving a term to the other side will change the sign.

`sqrt(9y-196) = 7-14`

`sqrt(9y-196) = -7`

Take the square of both sides to get rid of the radical or square root.

`(sqrt(9y-196))^2 = (-7)^2`

`9y - 196 = 49`

Combine similar terms.

`9y = 49 +196`

`9y = 245`

Divide both sides by the number beside y to leave y alone at the left side.

`(9y)/9 = 245/9`

`y = 27.22222`

`` or you can just leave `245/9` alone.

Check your answer by substituting 245/9 into y in the original given.

`sqrt(9*245/9) + sqrt(196) = sqrt(49)`

Simplifying that using your calculator, you have 29.65248 on the left side while you have 7 at the right side.

You can conclude that the right side and the left side of the equation are not equal so therefore what you get for y is not a solution of the equation.

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`sqrt(x^2)=+-x`

To mention this in this problem is not  correct

When we have specially given plus sign only.

The solutions of the equation are  `245/9` and `637/9`  given by one educator.Let verify its validty in given equation (fortunately you can not change equation)

`sqrt(9y-196)+sqrt(196)=sqrt(49)`

`y=245/9`

`sqrt(9xx(245/9)-196)+sqrt(196)=sqrt(49)`

`sqrt(245-196)+sqrt(196)=sqrt(49)`

`sqrt(49)+sqrt(196)=sqrt(49)`

Is it correct ? Now How will you justify  answer ! No.

`y=637/9`

`sqrt(9xx(637/9)-196)+sqrt(196)=sqrt(49)`

`sqrt(637-196)+sqrt(196)=sqrt(49)`

`sqrt(441)+sqrt(196)=sqrt(49)`

Is is correct  ? Again can you justify answer ! Never

This equation has no real root at all . If talk about complex root then it may possible.

`sqrt(9y-196)+sqrt(196)=sqrt(49)`

`sqrt(9y-196)+14=7`

`sqrt(9y-196)=7-14`

`sqrt(9y-196)=7i^2`

`9y=196+49i^4`

`9y=245`

`y=245/9`

But we have shown above `y=245/9`  does not satisfy the given equation.

Thus this equation has neither real root nor complex root.

So logic is simple

Nonnegative real number never equal to negative number.

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`sqrt(9y-196)+sqrt(196)=sqrt(49)`

`sqrt(9y-196)+-14=+-7`

`sqrt(9y-196)=+-7+-14`

`sqrt(9y-196)= +-21`  `sqrt(9y-196)=+-7`

`9y-196=441`     `9y-196= 49`

`9y= 637`               `9y= 245`

`y=637/9`                    `y=245/9`

Proof:

`y=637/9`

`sqrt(637-196)+sqrt(196)=sqrt(49)`

`sqrt(441)+sqrt(196)=sqrt(49)`

`+-21+-14=+-7`

That is:

`21-14=7`   and   `-21+14=-7`

proof `637/9`   OK.

Proof `245/9` :

`sqrt(245-196)+sqrt(196)=sqrt(49)`

`sqrt(49)+sqrt(196)=sqrt(49)`

`+-7+-14=+-7`

That is :

`-7+14=7`          `7-14=-7`

proof `245/9`   OK.

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We  have given

`sqrt(9y-196)=sqrt(49)-sqrt(196)`

`sqrt(9y-196)=7-14`

`sqrt(9y-196)=-7`

which is not possible , because left hand side is positive and right hand side is negative.

Thus  this equation has no root.