Problem 60 - Algebra 2

Question

What is the solution of \(x+\frac{1}{x}=-2 ?\) F. 1 and \(-1\) G. 0 only H. \(-\frac{1}{2}\) only J. \(-1\) only

Step-by-Step Solution

Answer

J. \(-1\) only

Step 1: Simplifying the Equation

The aim is to get rid of the fraction in the equation and make it easy to solve. This can be done by multiplying each term in the equation by x. This results in: \(x^2+1=-2x\)

Step 2: Rearrange the Equation

The intention is getting to the simplified form of a quadratic equation, i.e., \(ax^2 + bx + c = 0\). We can rearrange the equation from Step 1 to get: \(x^2 +2x+1= 0\)

Step 3: Identify the Quadratic Equation

At this point, the equation \(x^2 +2x+1= 0\) is identical to \((x + 1)^2 = 0\). This form is much easier to solve as it's basically a perfect square equation.

Step 4: Solve the Quadratic Equation

In the equation \((x + 1)^2 = 0\), the solution is found by solving for x, i.e \(x + 1 = 0\). By subtracting 1 from both sides, x equals -1

Key Concepts

Solving Equations

Solving equations is a fundamental part of mathematics that involves finding the values of variables that make the equation true. For the problem given, we first encounter a mixture of linear terms and fractions. This exercise involves manipulating the equation to find solutions for the variable \( x \).

To simplify equations with fractions, it's often easiest to eliminate them by multiplying through by the denominator. In the equation \( x + \frac{1}{x} = -2 \), multiplying every term by \( x \) eliminates the fraction, resulting in \( x^2 + 1 = -2x \). This is an important step because it transforms the equation from a fraction-dominated one to a polynomial form that is much easier to work with.

From here, you rearrange the terms to obtain a standard quadratic form \( ax^2 + bx + c = 0 \), such as \( x^2 + 2x + 1 = 0 \). This step paves the way for solving the quadratic equation using various methods such as factoring, applying the quadratic formula, or by recognizing patterns like a perfect square trinomial.

Factoring

Factoring is a technique used to simplify polynomials by expressing them as a product of their factors. This is particularly useful in solving quadratic equations.

In the equation \( x^2 + 2x + 1 = 0 \), you recognize it by looking for pairs of numbers that multiply to give you the constant term (1 in this case) and add up to give the linear coefficient (2 in this case). Notice how this fits into a pattern, which we'll describe fully later in the section on 'Perfect Square Trinomial'.

By identifying this, you can write the quadratic expression \( x^2 + 2x + 1 \) as \((x+1)(x+1)\) or more succinctly as \((x+1)^2\). This transformation shows that there is essentially only one unique solution to the equation, as it squared one of the factors to equal zero.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be factored into the square of a binomial. These trinomials are of the form \( a^2 + 2ab + b^2 = (a + b)^2 \). Recognizing this pattern allows for quick factoring and solving of the equation.

The equation from the problem, \( x^2 + 2x + 1 = 0 \), fits this pattern perfectly. Here, \( a = x \) and \( b = 1 \), so \( a^2 + 2ab + b^2 = (x + 1)^2 \).

• The first term \( x^2 \) is \( a^2 \).
• The middle term \( 2x \) comes from \( 2ab \).
• The last term \( 1 \) is \( b^2 \).

Recognizing a perfect square trinomial simplifies solving since you set \( (x + 1)^2 = 0 \), leading directly to the solution \( x = -1 \). This understanding is powerful in solving and interpreting quadratic functions efficiently.