How Do You Solve This Complex Irrational Equation?

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In summary, an irrational equation contains at least one irrational number such as pi, square root of 2, or e, and cannot be expressed as a ratio of two integers. This is different from a rational equation, which contains only rational numbers. Solving irrational equations is important in real-world situations and helps us understand the relationships between rational and irrational numbers. The steps to solve an irrational equation involve isolating the variable, simplifying, and squaring both sides. Special considerations include checking for extraneous solutions and being aware of restrictions on the domain of the variable.
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Solve the irrational equation $x^4-9x^3+16x^2+15x+26=\dfrac{7}{\sqrt{x^2-10x+26}+\sqrt{x^2-10x+29}+\sqrt{x^2-10x+41}}$.
 
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My solution:

With a bit of factoring, we obtain:

\(\displaystyle (x-5)^4+11(x-5)^3+31(x-5)^2+1=\frac{7}{\sqrt{(x-5)^2+1}+\sqrt{(x-5)^2+4}+\sqrt{(x-5)^2+16}}\)

So, let $u=x-5$, and we have:

\(\displaystyle u^4+11u^3+31u^2+1=\frac{7}{\sqrt{u^2+1}+\sqrt{u^2+4}+\sqrt{u^2+16}}\)

Define:

\(\displaystyle f(u)=u^4+11u^3+31u^2+1\)

Hence:

\(\displaystyle f'(0)=4u^3+33u^2+62u=u\left(4u^2+33u+62\right)\)

Using the first derivative test, we then find a local minimum at:

\(\displaystyle u=-\frac{33+\sqrt{97}}{8}\)

A local maximum at:

\(\displaystyle u=-\frac{33-\sqrt{97}}{8}\)

And a local minimum at:

\(\displaystyle u=0\)

From this, we obtain:

\(\displaystyle f_{\min}=f(0)=1\)

Now, we we define:

\(\displaystyle g(u)=\frac{7}{\sqrt{u^2+1}+\sqrt{u^2+4}+\sqrt{u^2+16}}\)

It is easy to see that:

\(\displaystyle g_{\max}=g(0)=1\)

Hence, the only solution is:

\(\displaystyle u=0\implies x=5\)
 
  • #3
Aww...that solution is superb! Well done MarkFL!

But I suspect it was Santa who told you how to do it, since you solved it so fast, hehehe...ho ho ho!:p
 

FAQ: How Do You Solve This Complex Irrational Equation?

What is an irrational equation?

An irrational equation is an equation that contains an irrational number, which is a number that cannot be expressed as a ratio of two integers. Irrational numbers include numbers like pi, square root of 2, and e.

How is an irrational equation different from a rational equation?

A rational equation contains only rational numbers, which can be expressed as a ratio of two integers. Irrational equations, on the other hand, contain at least one irrational number.

Why do we need to solve irrational equations?

Irrational equations often arise in real-world situations, and solving them allows us to find the exact solutions and make accurate calculations. They also help us understand the relationships between rational and irrational numbers.

What are the steps to solve an irrational equation?

The steps to solve an irrational equation are similar to solving a rational equation. You need to isolate the variable on one side of the equation and simplify the other side. Then, square both sides of the equation to eliminate the radical and solve for the variable.

Are there any special considerations when solving irrational equations?

Yes, when solving irrational equations, you need to be careful to check for extraneous solutions. These are solutions that may satisfy the equation mathematically but do not make sense in the context of the problem. You also need to be aware of any restrictions on the domain of the variable.

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