a) $\displaystyle \sqrt{2x+1} - \sqrt{x} = 1 $
True or false?
b) $\displaystyle \sqrt{x+2} + \sqrt{7x+2} = 6 $
True or false?
Can you please help me understand this.
a) $\displaystyle \sqrt{2x+1} - \sqrt{x} = 1 $
True or false?
b) $\displaystyle \sqrt{x+2} + \sqrt{7x+2} = 6 $
True or false?
Can you please help me understand this.
a) $\displaystyle \sqrt{2x+1} - \sqrt{x} = 1 $
True or false?
$\displaystyle \sqrt{2x+1}=(1+\sqrt{x})$
Square both sides...
$\displaystyle 2x+1=1+2\sqrt{x}+x$
Subtract $\displaystyle 1$ and $\displaystyle x$ from the right...
$\displaystyle x=2\sqrt{x}$
We see that $\displaystyle x=0$
Plugging it in shows that $\displaystyle \sqrt{0+1}-\sqrt{0}=1$
So this is a true statement.
You can try B...
Here's B anyway since it's more involved....
$\displaystyle \sqrt{x+2} + \sqrt{7x+2} = 6 $
$\displaystyle (\sqrt{x+2})=(6-\sqrt{7x+2})$
Square both sides...
$\displaystyle x+2=36-12\sqrt{7x+2}+7x+2$
Subtract 36, 7x, and 2 from the right.
$\displaystyle -6x-36=-12\sqrt{7x+2}$
Look, we can divide the equation by -6.
$\displaystyle (x+6)=(2\sqrt{7x+2})$
Square both sides again...
$\displaystyle x^2+12x+36=4(7x+2)$
Distribute the 4.
$\displaystyle x^2+12x+36=28x+8$
Subtract 28x and 8 from the right.
$\displaystyle x^2-16x+28=0$
This factors to...
$\displaystyle (x-14)(x-2)$
So our solutions are 14 and 2. Check them! You will find that 2 is the only solution and thus the statement is true.
Were those really "true or false" questions? I would interpret "true or false" for an equation involving, say, x, to be asking if it was true for all x- and that's clearly false for both questions.
To solve equations involving $\displaystyle \sqrt{x}$, you square to get rid of the square roots. After getting to $\displaystyle x= 2\sqrt{x}$, square again to get $\displaystyle x^2= 4x$ which is the same as $\displaystyle x^2- 4x= x(x- 4)= 0$. Either x= 0 or x- 4= 0 so x= 4.
Since squaring both sides like that can introduce "extraneous roots", you should always check by putting the numbers into the original equation. The original equation was [tex]\sqrt{2x+1}- \sqrt{x}= 1[tex]. With x= 0 that becomes $\displaystyle \sqrt{2(0)+1}- \sqrt{0}= \sqrt{1}= 1$ so x= 0 is a solution. With x= 4, it becomes $\displaystyle \sqrt{2(4)+ 1}- \sqrt{4}= \sqrt{9}- \sqrt{4}= 3- 2= 1$ so x= 4 is also a solution.