91Ó°ÊÓ

The substitution method is a helpful approach for solving equations by reducing their complexity. In this problem, we faced an equation with negative exponents: \[ 2 x^{-2} - 3 x^{-1} - 4 = 0. \]
To make it easier, we can introduce a new variable. We set \[ y = x^{-1}. \]This step simplifies our equation by transforming it into a more manageable quadratic form: \[ 2 y^2 - 3 y - 4 = 0. \]By substituting back the new variable later, we can find the solutions for the original variable without much hassle. The substitution method is beneficial when dealing with equations featuring complex exponents, fractions, or other expressions that might initially seem challenging to solve.

The quadratic formula is a general solution for quadratic equations. These equations take the form: \[ a y^2 + b y + c = 0, \]where \(a\), \(b\), and \(c\) are constants. To solve the quadratic equation: \[ 2 y^2 - 3 y - 4 = 0, \]we use the quadratic formula: \[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]
In our case:

• \( a = 2 \)
• \( b = -3 \)
• \( c = -4 \)

Substituting these values, we find: \[ y = \frac{3 \pm \sqrt{9 + 32}}{4} = \frac{3 \pm \sqrt{41}}{4}. \]This provides us with two potential solutions for \(y\). Understanding the quadratic formula is crucial since it offers a straightforward way to resolve any quadratic equation efficiently.

After solving the quadratic equation, we needed to back-substitute \( y = x^{-1} \) to find \( x \). This led to the solutions: \[ x_1 = \left( \frac{3 + \sqrt{41}}{4} \right)^{-1} = \frac{4}{3 + \sqrt{41}} \] and \[ x_2 = \left( \frac{3 - \sqrt{41}}{4} \right)^{-1} = \frac{4}{3 - \sqrt{41}}. \]
These expressions can be further simplified by rationalizing the denominators. Rationalizing involves getting rid of the radical in the denominator by multiplying both the numerator and denominator by the conjugate of the denominator. For instance: \[ \frac{4}{3 + \sqrt{41}} \times \frac{3 - \sqrt{41}}{3 - \sqrt{41}} = \frac{4 (3 - \sqrt{41})}{ (3 + \sqrt{41})(3 - \sqrt{41})} = \frac{4 (3 - \sqrt{41})}{3^2 - (\sqrt{41})^2} = \frac{4 (3 - \sqrt{41})}{9 - 41} = \frac{4 (3 - \sqrt{41})}{-32} = -\frac{3 - \sqrt{41}}{8}. \]The process cleans up the solution, making it more elegant and easier to understand. Rationalizing denominators is an important skill, as it often simplifies and clarifies final expressions in algebra.