How to Solve an Equation with Square Roots?

In summary, the conversation revolved around solving the equation \sqrt{x}+\sqrt{x+8}=8. The expert summarizer provided steps to rearrange the equation and correctly square both sides to find the solution x=28. The conversation ended with the confirmation that the solution works in the original equation.
  • #1
magda21
3
0
Please Help me solve it \[ \sqrt{x}+\sqrt{x+8}=8 \] thanks
 
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  • #2
Hello, and welcome to MHB! :)

I would begin by arranging as follows:

\(\displaystyle \sqrt{x+8}=8-\sqrt{x}\)

I really just want to arrange it so that there isn't two radicals on the same side. Now, what do you get when you square both sides?
 
  • #3
x+8=64-x
2x=56
x=28
what I'm doing wrong?
 
  • #4
Ah, you are not squaring the RHS correctly. Let's go back to:

\(\displaystyle \sqrt{x+8}=8-\sqrt{x}\)

Now, recall that:

\(\displaystyle (a+b)^2=a^2+2ab+b^2\)

And so, when we square both sides of our equation (bearing in mind that we must check for extraneous solutions), we get:

\(\displaystyle x+8=64-16\sqrt{x}+x\)

Collecting like terms, we can arrange this as:

\(\displaystyle 16\sqrt{x}=56\)

Divide through by 8:

\(\displaystyle 2\sqrt{x}=7\)

Can you proceed?
 
  • #5
\[ \sqrt{x}=\frac{7}{2} \]
x=\frac{49}{4}
I see, thank you so much
 
  • #6
Yes, and once we verifiy is works in the original equation, which it does, then we're done. :)
 

1. What is a square root?

A square root is a mathematical operation that finds the number which, when multiplied by itself, gives the original number. It is represented by the symbol √, and the number under the symbol is called the radicand.

2. How do you solve an equation with a square root?

To solve an equation with a square root, you need to isolate the square root on one side of the equation and square both sides. This will eliminate the square root and allow you to solve for the variable.

3. Can an equation with a square root have more than one solution?

Yes, an equation with a square root can have more than one solution. This is because when you square both sides of the equation, you introduce the possibility of extraneous solutions, which are solutions that do not actually satisfy the original equation.

4. Can an equation with a square root have a negative solution?

Yes, an equation with a square root can have a negative solution. This is because when you square both sides of the equation, you eliminate the square root and the resulting equation may have both positive and negative solutions.

5. Are there any special rules for solving equations with square roots?

Yes, there are a few special rules for solving equations with square roots. These include the rule that if both sides of the equation are squared, the resulting equation may have extraneous solutions, and the rule that when solving for a variable, you must always check your answer by plugging it back into the original equation.

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