Problem 21 - Algebra and Trigonometry

Question

1–54 ? Find all real solutions of the equation. $$ \frac{1}{x-1}-\frac{2}{x^{2}}=0 $$

Step-by-Step Solution

Answer

There are no real solutions to the equation.

Step 1: Understand the Equation

The equation \( \frac{1}{x-1}-\frac{2}{x^{2}}=0 \) is composed of two rational expressions. To solve for \( x \), we need to find a common denominator to combine these expressions.

Step 2: Find Common Denominator

The common denominator for the expression is \( x^2(x-1) \). We multiply each term by this common denominator to eliminate the fractions.

Step 3: Eliminate Fractions

Multiply each term by the common denominator \( x^2(x-1) \): - \( \frac{1}{x-1} \times x^2(x-1) = x^2 \) - \( \frac{2}{x^2} \times x^2(x-1) = 2(x-1) \) This simplifies our equation to \( x^2 - 2(x-1) = 0 \).

Step 4: Simplify the Equation

Distribute and simplify the equation: \( x^2 - 2x + 2 = 0 \).

Step 5: Solve the Quadratic Equation

This is a quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -2 \), and \( c = 2 \). The solutions for \( x \) can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values:- \( b^2 - 4ac = (-2)^2 - 4 \times 1 \times 2 = 4 - 8 = -4 \).

Step 6: Interpret the Results

Since the discriminant \( b^2 - 4ac \) is negative, \( -4 \), there are no real solutions to this equation. The solutions would be complex.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations. A quadratic equation is any equation that can be transformed into the standard form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. To solve these, we apply the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula helps find the roots of a quadratic equation by calculating values of \( x \) that would satisfy the equation. It accounts for all possible solutions, including real and complex numbers.
In the solution provided, we applied the quadratic formula to \( x^2 - 2x + 2 = 0 \) by identifying \( a = 1 \), \( b = -2 \), and \( c = 2 \).

Common Denominator

When dealing with rational expressions such as fractions, finding a common denominator is crucial. A common denominator is a shared multiple of the denominators of each fraction involved. To combine fractions, we adjust each term to have this common denominator, allowing us to add or subtract them effectively without modifying their actual values.
In this problem, the common denominator was determined to be \( x^2(x-1) \). We multiplied each term by this shared denominator to maintain the equation's equality while eliminating the fractions. This technique simplified the expression from a rational equation to a solvable quadratic form.

Rational Expressions

Rational expressions are fractions where the numerator and/or the denominator are polynomials. They play a crucial role in various mathematical equations, particularly when working with algebraic fractions. The key to handling rational expressions is to simplify them when possible and to find common denominators to combine different parts of the expression.
The equation \( \frac{1}{x-1}-\frac{2}{x^{2}}=0 \) consists of these types of expressions. By understanding how to manipulate them, such as finding common denominators, we can transform a complex equation into a much simpler form, as seen in the steps where fractions were removed and the equation was rewritten.

Discriminant

The discriminant is a critical part of the quadratic formula under the square root: \( b^2 - 4ac \). It determines the nature and number of solutions that a quadratic equation has.

• If the discriminant is positive, the equation has two distinct real solutions.
• If it is zero, there is exactly one real solution (a repeated root).
• If negative, the quadratic equation has no real solutions but rather two complex solutions.

In the exercise, we calculated the discriminant as \( (-2)^2 - 4 \times 1 \times 2 = -4 \). A negative discriminant means the equation has no real solutions, conveying that any potential solutions are complex numbers, involving imaginary units.