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The substitution method is a powerful tool in solving complex equations. The central idea is to replace a complicated part of the equation with a simpler substitute, making solving the problem more manageable. In this exercise, we substituted a part of the given equation to make it simpler.
When trying to solve the equation \( x^{1 / 2} + 3 x^{-1 / 2} = 10 x^{-3 / 2} \), we noticed that simplifying this might be easier if expressed in terms of another variable.
Thus, by setting \( y = x^{-1/2} \), we transform the equation into a simpler form \( \frac{1}{y} + 3y = 10y^3 \), which is more straightforward to manipulate and interpret.
This substitution reduces the problem to a polynomial-like form, where traditional algebraic techniques, such as clearing fractions and factoring, can be applied.

Polynomial factoring is a technique used to break down a polynomial into simpler, more manageable components called factors. Once a polynomial is factored, finding solutions becomes much easier.
In our example, the equation \( 10y^4 - 3y^2 - 1 = 0 \) was reached by simplifying and rearranging the substituted variables.
Recognizing it as a quadratic in form, we performed another substitution with \( z = y^2 \).

• This substitution transformed the equation further into \( 10z^2 - 3z - 1 = 0 \).
• We factored this expression as \( (5z + 1)(2z - 1) = 0 \).

Through this process, the polynomial was reduced to its essential components, making it simple to set each factor to zero and solve the equation.

Variable substitution plays a key role in simplifying and solving complex mathematics problems. This method essentially replaces complex terms within an equation with simpler ones, aiding in a more straightforward resolution.
Initially, we set \( y = x^{-1/2} \) to reduce complexity in the equation. After substitution and simplification, the variable \( z = y^2 \) was introduced. Each substitution was designed strategically to transform the problem into something more solvable.
This approach utilized two levels of substitution:

• Turning complicated power expressions into simpler fractional forms, and later
• Transforming a quartic polynomial into a quadratic one.

Such techniques are beneficial, especially for making large, seemingly daunting problems into more manageable pieces, eventually leading to solutions for the original variables.

Real solutions are the real-number results that satisfy an equation. In the context of our problem, we sought real solutions to the substituted equation.
Upon factoring the polynomial \( 10z^2 - 3z - 1 = 0 \), we found potential solutions \( z = -\frac{1}{5} \) and \( z = \frac{1}{2} \). Considering \( z = y^2 \), and knowing that square values must be non-negative, we found that \( z = -\frac{1}{5} \) isn't possible.
Therefore, we focused on \( y^2 = \frac{1}{2} \) leading to two possibilities: \( y = \frac{1}{\sqrt{2}} \) and \( y = -\frac{1}{\sqrt{2}} \).

• Reverting to the original variable, we found the real solution \( x = 2 \) by calculating \( x = (\sqrt{2})^2 = 2 \).

Negative results for \( y \) were disregarded in terms of \( x \), as squaring implies non-negative results, affirming the real, positive nature of the solution.