Can the Algebraic Expression $8-2\sqrt7$ be Simplified to a Perfect Square?

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In summary, by setting the given expression equal to a perfect square and solving for the variables, it is shown that $8-2\sqrt7$ is a perfect square number. Taking the square root on both sides would result in a tedious algebraic process.
  • #1
mathdad
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See picture. Show that the RHS = LHS without using a calculator.

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  • #2
Well, let's see...

Is $8-2\sqrt7$ a perfect square number?

$$(a+b\sqrt7)^2=a^2+2ab\sqrt7+7b^2$$

$$a^2+7b^2=8$$

$$2ab=-2$$

$$\implies a=-1,b=1$$

Hence $8-2\sqrt7=(\sqrt7-1)^2$

$$\sqrt7-\sqrt{8-2\sqrt7}=\sqrt7-(\sqrt7-1)=\sqrt7-\sqrt7+1=1$$
 
  • #3
greg1313 said:
Well, let's see...

Is $8-2\sqrt7$ a perfect square number?

$$(a+b\sqrt7)^2=a^2+2ab\sqrt7+7b$$
Check the last term.
\(\displaystyle (a+b\sqrt7)^2=a^2+2ab\sqrt7+7b^2\)

-Dan
 
  • #4
topsquark said:
Check the last term.

Post corrected. Thanks Dan.
 
  • #5
greg1313 said:
Well, let's see...

Is $8-2\sqrt7$ a perfect square number?

$$(a+b\sqrt7)^2=a^2+2ab\sqrt7+7b^2$$

$$a^2+7b^2=8$$

$$2ab=-2$$

$$\implies a=-1,b=1$$

Hence $8-2\sqrt7=(\sqrt7-1)^2$

$$\sqrt7-\sqrt{8-2\sqrt7}=\sqrt7-(\sqrt7-1)=\sqrt7-\sqrt7+1=1$$

What would the algebra look like if I take the square root on both sides? Is it tedious?
 

FAQ: Can the Algebraic Expression $8-2\sqrt7$ be Simplified to a Perfect Square?

What is a radical equation?

A radical equation is an equation that contains a radical, or a square root symbol. It is used to solve for a variable that is under the radical.

How do you solve a radical equation?

To solve a radical equation, you must first isolate the radical on one side of the equation. Then, square both sides of the equation to eliminate the radical. Finally, solve for the variable and check your solution by plugging it back into the original equation.

What are extraneous solutions in radical equations?

Extraneous solutions are solutions that arise during the process of solving a radical equation but do not actually satisfy the original equation. They can occur when squaring both sides of an equation to eliminate a radical, as this can introduce additional solutions that do not work in the original equation.

What is the difference between a radical equation and a quadratic equation?

A radical equation contains a radical, while a quadratic equation contains a squared variable. Both equations can be solved using similar methods, but the solutions for a radical equation may need to be checked for extraneous solutions. Additionally, a quadratic equation can have multiple solutions, while a radical equation typically only has one.

Can radical equations have imaginary solutions?

Yes, radical equations can have imaginary solutions. This can occur when taking the square root of a negative number, which results in an imaginary number. Imaginary solutions are represented by the letter "i" and can be used to solve equations, but they are not real numbers.

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