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anemone
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Find the largest positive real solution to the equation $7x\sqrt{x+1}-3=2x^2+3x$.
Opalg said:[sp]$7x\sqrt{x+1} = 2x^2+3x + 3$. Square both sides: $49x^2(x+1) = \bigl(2x^2+3x + 3\bigr)^2 = 4x^4 + 12x^3 + 21x^2 + 18x + 9$, so that $4x^4 - 37x^3 - 28x^2 + 18x + 9 = 0.$ That factorises as $\bigl(x^2 - 9x - 9\bigr)\bigl(4x^2 - x - 1\bigr) = 0.$ The positive roots are $\frac12\bigl(9+3\sqrt{13}\bigr)$ and $\frac18\bigl(1+ \sqrt{17}\bigr)$. The first of those is the larger. But it came from squaring the original equation, so we have to check that it satisfies that equation and was not introduced by squaring. It does satisfy the original equation, so the answer is $\frac12\bigl(9+3\sqrt{13}\bigr) \approx 9.908.$[/sp]
A positive real root is a value that, when substituted into an equation, satisfies the equation and is greater than zero.
Finding the largest positive real root can be useful in many applications, such as in economics, engineering, and physics, as it can help determine the maximum value or point of stability for a system.
Some common methods for finding the largest positive real root include using the quadratic formula, factoring, and graphing the equation.
Yes, there are some limitations to finding the largest positive real root. For example, some equations may not have real roots, or there may be multiple positive real roots. Additionally, some methods may not work for certain types of equations.
The largest positive real root can be used in practical applications to determine the maximum value or point of stability for a system, to find the optimal solution for a problem, or to analyze the behavior of a system over time.