# Thread: Simple equation help

1. ## Simple equation help

Hi, im having problems with a simple equation that looks like this:
$\displaystyle \sqrt{x+3} = \sqrt{x-2} + \sqrt{x-5}$

After squaring both sides i have: $\displaystyle x+3=2x-7 +2\sqrt{x-2}\cdot \sqrt{x-5} \rightarrow x-10+2 \cdot (x-2)^\frac{1}{2} \cdot (x-5) ^\frac {1}{2}$ Wich gives: $\displaystyle x-10+2x^{2}-14x+20=0$

However, the anser should be 6 and i get 4. Im not sure about what i am doing wrong =/

2. When you multiplied sqrt(x - 2) and sqrt(x - 5) to each other, you forgot to include the square root sign in the product.

You ended up with a final expression that doesn't include any square root signs so it became inaccurate.

3. Originally Posted by Jones
Hi, im having problems with a simple equation that looks like this:
$\displaystyle \sqrt{x+3} = \sqrt{x-2} + \sqrt{x-5}$

After squaring both sides i have: $\displaystyle x+3=2x-7 +2\sqrt{x-2}\cdot \sqrt{x-5} \rightarrow x-10+2 \cdot (x-2)^\frac{1}{2} \cdot (x-5) ^\frac {1}{2}$ Wich gives: $\displaystyle x-10+2x^{2}-14x+20=0$

However, the anser should be 6 and i get 4. Im not sure about what i am doing wrong =/
Starting from this step:

$\displaystyle x-10+2 \cdot (x-2)^\frac{1}{2} \cdot (x-5) ^\frac {1}{2}=0$

This becomes: $\displaystyle \sqrt{x^2-7x+10}=\tfrac{1}{2}(10-x)$

Now square both sides:

$\displaystyle x^2-7x+10=\tfrac{1}{4}(x^2-20x+100)$

$\displaystyle \implies 4x^2-28x+40=x^2-20x+100$

$\displaystyle \implies 3x^2-8x-60=0$

$\displaystyle \implies (x-6)(3x+10)=0$

$\displaystyle \implies \color{red}\boxed{x=6} \ \text{or} \ \color{red}\boxed{x=-\tfrac{10}{3}}$

Test values:

$\displaystyle \sqrt{-\tfrac{10}{3}+3} = \sqrt{-\tfrac{10}{3}-2} + \sqrt{-\tfrac{10}{3}-5}$

$\displaystyle \sqrt{\tfrac{1}{3}}i=4\sqrt{\tfrac{1}{3}}i+5\sqrt{ \tfrac{1}{3}}i$

This doesn't work...

$\displaystyle \sqrt{6+3} = \sqrt{6-2} + \sqrt{6-5}$

$\displaystyle \implies 3=2+1 \implies 3=3$

$\displaystyle x=6$ is the only solution.

4. Originally Posted by Serena's Girl
When you multiplied sqrt(x - 2) and sqrt(x - 5) to each other, you forgot to include the square root sign in the product.

You ended up with a final expression that doesn't include any square root signs so it became inaccurate.
Since we have $\displaystyle \sqrt(x-2) \cdot \sqrt(x-5)$
Its the same thing as $\displaystyle (x-2)^\frac{1}{2} \cdot (x-5)^\frac{1}{2}$

Accordign to the exponetial rules $\displaystyle A^b \cdot B^a = AB^{a + b}$ and $\displaystyle \frac{1}{2} + \frac {1}{2} = 1$

Therefore $\displaystyle ((x-2) \cdot (x-5))^{1}$

5. Originally Posted by Jones
Since we have $\displaystyle \sqrt(x-2) \cdot \sqrt(x-5)$
Its the same thing as $\displaystyle (x-2)^\frac{1}{2} \cdot (x-5)^\frac{1}{2}$

Accordign to the exponetial rules $\displaystyle A^b \cdot B^a = AB^{a + b}$ and $\displaystyle \frac{1}{2} + \frac {1}{2} = 1$

Therefore $\displaystyle ((x-2) \cdot (x-5))^{1}$
But you can only do that if the exponential functions have the same base.

In the case of square roots, $\displaystyle \sqrt{a}\cdot\sqrt{b}=\sqrt{ab} \ \forall a\geq 0, \ b\geq 0$

--Chris

6. Oh, right.

Thank you