How can I solve radical equations using the given hint?

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In summary, to determine all of the real number solutions for the radical equation sqrt{x^4 - 13x^2 + 37} = 1, you can use the hint given in the textbook to let x^2 = t and x^4 = t^2. Then proceed as usual by squaring both sides, subtracting 1 from both sides, factoring and substituting back in. It is important to always check the answer(s) when dealing with radical equations to ensure that they are valid solutions.
  • #1
mathdad
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Determine all of the real number solutions for the radical equation.

sqrt {x^4 - 13x^2 + 37} = 1

Hint given in textbook:

Let x^2 = t and x^4 = t^2

After applying the hint given, do I proceed as usual by squaring both sides?

I have to back-substitute somewhere in the solution steps, right?

Why is it important to always check the answer (s) when dealing with radical equations?
 
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  • #2
Yes, it is solving as usual.
You can square both sides, then subtract 1 from both sides.
Factor as usual. Substitute back in.

Try it out and see if you get the correct answers.
 
  • #3
joypav said:
Yes, it is solving as usual.
You can square both sides, then subtract 1 from both sides.
Factor as usual. Substitute back in.

Try it out and see if you get the correct answers.

Great. Ok. It's not so bad.
 

FAQ: How can I solve radical equations using the given hint?

What is a radical equation?

A radical equation is an equation that contains a square root or other radical symbol. It is also known as an equation with a radical expression.

How do you solve a radical equation?

To solve a radical equation, you must isolate the radical expression and then square both sides of the equation. This will eliminate the radical and allow you to solve for the variable.

What are some common mistakes when solving radical equations?

Some common mistakes when solving radical equations include forgetting to square both sides of the equation, incorrectly distributing a minus sign, and not checking for extraneous solutions.

Can a radical equation have more than one solution?

Yes, a radical equation can have more than one solution. This is because when the radical is eliminated by squaring both sides, it can result in two different values for the variable.

How are radical equations used in real life?

Radical equations are used in real life to solve problems involving distance, time, and other quantities that can be represented by square roots. They are also used in various fields of science, such as physics and engineering, to model and solve complex problems.

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