Solve for x and y When (x+$\sqrt {x^2+1})\times (y+\sqrt {y^2+4})=7$

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In summary, the equation (x+$\sqrt {x^2+1})\times (y+\sqrt {y^2+4})=7$ can be solved for an infinite number of solutions and can be solved by hand using algebraic manipulation. The values for x and y can be any real numbers, but some may result in imaginary solutions. This equation has practical applications in fields such as physics, engineering, and economics for solving unknown variables involving square roots.
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Albert1
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$if \,\, (x+\sqrt {x^2+1})\times (y+\sqrt {y^2+4})=7$

$find: \,\, x\sqrt {y^2+4}+ y\sqrt {x^2+1}=?$
 
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My attempt:
Let $x + \sqrt{x^2+1}$ take on the value $\alpha$ (note that $\alpha > 0$ for all $x$) and solve the given equation for $y$: \[\alpha (y+\sqrt{y^2+4})=7 \Rightarrow y = \frac{7}{2\alpha }-\frac{2\alpha }{7}\]In order to facilitate the algebra let $\beta = \frac{7}{2\alpha } > 0$, for all $x$.We are looking for the value of the sum: $x\sqrt{y^2+4}+y\sqrt{x^2+1}$. Elaborating on each term:(i). \[x\sqrt{y^2+4} = x\sqrt{\left ( \beta -\frac{1}{\beta } \right )^2+4}=x\sqrt{\left ( \beta +\frac{1}{\beta } \right )^2}= x\left ( \beta +\frac{1}{\beta } \right )\]
(ii). \[y\sqrt{x^2+1} = y(\alpha -x)=\left (\beta -\frac{1}{\beta } \right )\left ( \alpha -x \right )\]Summing the terms:

\[x\sqrt{y^2+4}+y\sqrt{x^2+1} =x\left ( \beta +\frac{1}{\beta} \right )+\left (\beta -\frac{1}{\beta} \right )\left ( \alpha -x \right )\]\[=\alpha \beta +\frac{1}{\beta }\left ( 2x-\alpha \right )= \frac{7}{2}+\frac{2}{7}(x+\sqrt{x^2+1})(x-\sqrt{x^2+1}) = \frac{7}{2}-\frac{2}{7} = \frac{45}{14}.\]
 

FAQ: Solve for x and y When (x+$\sqrt {x^2+1})\times (y+\sqrt {y^2+4})=7$

What is the equation to solve for x and y?

The equation is (x+$\sqrt {x^2+1})\times (y+\sqrt {y^2+4})=7$

How many solutions are there for this equation?

There are an infinite number of solutions for this equation.

Can this equation be solved by hand?

Yes, this equation can be solved by hand using algebraic manipulation and solving for both x and y.

What are the possible values for x and y?

The values for x and y can be any real numbers that satisfy the equation. However, some values may result in imaginary solutions.

How can this equation be applied in real life situations?

This equation can be used in various fields such as physics, engineering, and economics to solve for unknown variables in equations involving square roots.

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