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- Feb 5, 2013

- 1,384

Rearrange:

$$\sqrt{x+15}=15-\sqrt{x}$$

Square both sides:

$$x+15=225-30\sqrt{x}+x$$

Subtract $x$ from both sides:

$$15=225-30\sqrt{x}$$

Rearrange:

$$30\sqrt{x}=210$$

Divide both sides by 30:

$$\sqrt{x}=7$$

$$x=49$$

Does that help?

- Mar 31, 2013

- 1,322

alternativelyI have been away from my math pursuits for some time. I don't remember how to solve the following equation for x

sqrt(x + 15) + sqrt(x) = 15

Any suggestions are appreciated how to approach the solution for this equation.

$\sqrt{x + 15} + \sqrt{x} = 15\cdots(1)$ (given)

we know $(x+15)- x = 15$ (identity)

or $(\sqrt{x + 15} + \sqrt{x}))(\sqrt{x + 15}- \sqrt{x}) = 15\cdots(2)$

deviding (2) by (1)

$\sqrt{x + 15} - \sqrt{x} = 1\cdots(3)$

add (1) and (3) to get

$2\sqrt{x + 15} = 16$

or $\sqrt{x + 15} = 8$ or $x+15=64$ or $x=49$

- Feb 27, 2018

- 2

Could you help me understand where the 30 comes from in $$x+15=225-30\sqrt{x}+x$$

Rearrange:

$$\sqrt{x+15}=15-\sqrt{x}$$

Square both sides:

$$x+15=225-30\sqrt{x}+x$$

Subtract $x$ from both sides:

$$15=225-30\sqrt{x}$$

Rearrange:

$$30\sqrt{x}=210$$

Divide both sides by 30:

$$\sqrt{x}=7$$

$$x=49$$

Does that help?

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- Feb 5, 2013

- 1,384

$$(15-\sqrt x)^2=(15-\sqrt x)(15-\sqrt x)=15\cdot15-15\sqrt x-15\sqrt x+x=225-30\sqrt x+x$$

Does that help?

- Feb 27, 2018

- 2

Perfectly. Thank you!

$$(15-\sqrt x)^2=(15-\sqrt x)(15-\sqrt x)=15\cdot15-15\sqrt x-15\sqrt x+x=225-30\sqrt x+x$$

Does that help?

- Jan 30, 2018

- 537

Notice that this depends upon the fact that, for a and b

- May 22, 2018

- 1

It's easiest to simply start testing integers.

And they must have an integer square root. That makes it pretty easy.

Just guess!

And there's another numerical clue.

The difference between n squared and (n+1) squared = n + n +1

So 15 is the giveaway. 7 + 8 = 15

So 7 squared is 49.

You can do it All In Your Head. No algebra necessary. That algebra warps your mind.

Find the easy way!

In fact, if you look at this truth: "difference between n squared and (n+1) squared = n + n +1"

You'll get the answer in milliseconds.

- Jan 30, 2018

- 537

Because, as you have been told repeatedly here, the square of $\sqrt{x- 15}+ \sqrt{x}$ is NOT $x- 15+ x$.

To square $a+ b$ you multiply $(a+ b)(a+ b)$. Using the "distributive law", that is $a(a+ b)+ b(a+ b)= a^2+ ab+ ba+ b^2= a^2+ 2ab+ b^2$, NOT $a^2+b^2$.

If that is not sufficiently convincing, $(3+ 4)^2= 7^2= 49$ NOT $3^2+ 4^2= 9+ 16= 25$. It IS $3^2+ 2(3)(4)+ 4^2= 9+ 24+ 16= 25+ 24= 49$.

To square $a+ b$ you multiply $(a+ b)(a+ b)$. Using the "distributive law", that is $a(a+ b)+ b(a+ b)= a^2+ ab+ ba+ b^2= a^2+ 2ab+ b^2$, NOT $a^2+b^2$.

If that is not sufficiently convincing, $(3+ 4)^2= 7^2= 49$ NOT $3^2+ 4^2= 9+ 16= 25$. It IS $3^2+ 2(3)(4)+ 4^2= 9+ 24+ 16= 25+ 24= 49$.

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