91Ó°ÊÓ

The substitution method is a powerful tool for solving complex equations. By replacing variables with simpler expressions, you simplify the equation. In this case, we started with the equation:
\( 2x^{2/5} + 3x^{1/5} = -1 \).
This equation involves fractional exponents, which can be intimidating. Using substitution, we can simplify the expression. We let \( y = x^{1/5} \). This way, \( y^2 = x^{2/5} \). Now, our daunting original equation becomes a quadratic:

• \( 2y^2 + 3y = -1 \).

Sometimes, substitution can transform a complex task into a more manageable problem. By converting into a quadratic form, we've reached a more familiar territory and can use well-known methods to solve it.

Once the equation has been transformed into a quadratic, the quadratic formula becomes extremely useful. The standard form of a quadratic equation is

• \( ax^2 + bx + c = 0 \),

where \( a \), \( b \), and \( c \) are constants. To solve for \( x \), the quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
In our modified equation \( 2y^2 + 3y + 1 = 0 \), \( a = 2 \), \( b = 3 \), and \( c = 1 \).
This universal formula allows solving any quadratic equation by providing the solutions using the coefficients directly. It's a crucial tool in the mathematician's toolkit and eliminates guesswork, leading directly to the roots of the equation.

The discriminant is a key concept when working with quadratics and appears in the quadratic formula as \( b^2 - 4ac \). It reveals significant information about the nature of the solutions:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the roots are complex and not real.

In our exercise, the discriminant was calculated as follows:
\( 3^2 - 4(2)(1) = 9 - 8 = 1 \).
This positive value indicates two distinct real roots for the equation. Calculating the discriminant is a vital part of analyzing quadratic equations because it provides insights before fully solving the equation.

Exponents and radicals often appear in equations, adding a layer of complexity. The equation \( 2x^{2/5} + 3x^{1/5} = -1 \) is a perfect example. Exponents such as \( x^{1/5} \) and \( x^{2/5} \) signify roots, as fractional exponents are another way to represent radicals.

• \( x^{1/5} \) means the fifth root of \( x \).
• \( x^{2/5} \) can be understood as the square of the fifth root of \( x \).

Recognizing these forms allows us to transform them into simpler terms through substitution and then solve through standard quadratic methods. When back-substituting to find \( x \), remember to handle roots and powers carefully, such as raising both sides of the equation to necessary powers to eliminate radicals. Understanding these concepts helps in breaking down seemingly tough algebraic expressions into manageable parts.