Problem 47 - Algebra and Trigonometry

Question

Solve each equation in Exercises 41–60 by making an appropriate substitution. $$x^{-2}-x^{-1}-20=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x=\frac{1}{5}\) and \(x=-\frac{1}{4}\).

Step 1: Substitution

Substitute \(x^{-1} = y\), simplifying the equation to \(y^2-y-20 = 0\)

Step 2: Factorizing

Use factor multiplication and addendum method to further break down the equation. This method involves finding two numbers that multiply to give the quadratic coefficient (-20) and add to give the linear coefficient (-1). Those numbers are -5 and 4. This leads to \((y-5)(y+4) = 0\).

Step 3: Finding roots for y

Set each factor equal to zero and solve the resulting equations for \(y\), resulting in \(y=5\) or \(y=-4\).

Step 4: Substituting y back to \(x^{-1}\)

Now, substitute \(y\) back to \(x^{-1}\) to get \(x^{-1}=5\) or \(x^{-1}=-4\).

Step 5: Solving for x

Finally, solving for \(x\), we get \(x=\frac{1}{5}\) or \(x=-\frac{1}{4}\).

Key Concepts

Quadratic Equations

Quadratic equations are algebraic expressions that set a second-degree polynomial equal to zero and typically have the general form of \(ax^{2} + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). The term 'quadratic' comes from \(quad\) meaning square, because the variable gets squared (like \(x^{2}\)).

They are fundamental in algebra and have various solution methods, such as factoring, completing the square, using the quadratic formula, or graphing. For most students, solving a quadratic equation becomes more manageable when transformed into a simpler form. Substitute terms with exponents with variables to reduce the equation. That's where substitution method comes into play, which brings us to our next concept.

Substitution Method

The substitution method is a powerful technique in algebra, used to solve equations and simplify expressions. When you're faced with a complicated algebraic equation or expression, substitution can help you transform it into an easier problem. For instance, in the context of quadratic equations, if you have a term like \(x^{-1}\), which might seem daunting, you can replace it with a temporary variable, like \(y\), making the equation look simpler and more familiar.

Once the substitution is made, you proceed to solve the equation as if it were a regular quadratic equation. After finding the solution(s) for the new variable, you then substitute back to find the original unknown. This back and forth transformation is particularly helpful when dealing with higher degree or inverse terms.

Factorization

Factorization, in the realm of algebra, is a technique to break down numbers or expressions into a product of their factors—elements that, when multiplied together, give back the original number or expression. When applied to quadratic equations, it involves restructuring the quadratic expression \(ax^{2} + bx + c = 0\) into a product of binomials, such as \((y-5)(y+4) = 0\).

In the substitution method, factorization becomes a straightforward approach to find roots, as you are often left with an equation that is easier to factor. The goal is to identify two numbers that both add up to the linear coefficient and multiply to the constant term. Once the equation is fully factored, it can be set equal to zero, and each binomial can be solved for the unknown variable. It's a critical step in the process because it makes the transition from a single complex equation to simpler ones.

Algebraic Equations

Algebraic equations are equations that involve variables, constants, and arithmetic operations. Fundamentally, they are declarations that two algebraic expressions are equal. In the context of the quadratic formula, algebraic equations become a playground where we can apply various methods to find the values of the unknowns that satisfy the equalities.

The beauty of algebraic equations lies in the various strategies available for finding solutions—whether through substitution, factorization, graphing, or other methods. Developing a deep understanding of these strategies allows students to approach these equations with flexibility and select the method that best fits the problem at hand. From simple linear equations to more complex systems, mastering the art of solving algebraic equations is a pinnacle achievement in any student's journey through mathematics.