Problem 5 - Elementary Algebra

Question

Which of the following equations are quadratic equations? Answer "yes" or "no" to each equation. $$ \frac{2}{x}-5 x^{2}=6 x+4 $$

Step-by-Step Solution

Answer

Equation: $\frac{2}{x} - 5x^{2} - 6x - 4 = 0$ Answer: No, the given equation is not a quadratic equation.

Step 1: Rewrite the Equation

First, rewrite the given equation with the goal of isolating the dependent variable, x, on one side of the equation: $$ \frac{2}{x} - 5x^{2} - 6x - 4 = 0 $$

Step 2: Clear the Fraction

To clear the fraction in the equation, multiply all terms by x: $$ 2 - 5x^{3} - 6x^{2} - 4x = 0 $$

Step 3: Rearrange and Check if it is Quadratic

Rearrange the terms in descending order of powers of x: $$ -5x^{3} - 6x^2 - 4x + 2 = 0 $$ Notice that the highest power of x in the above equation is 3. Since the highest power of x in the equation is not 2, this equation is not a quadratic equation. So, the answer to the given question is "no".

Key Concepts

Polynomials

Polynomials are expressions consisting of variables and coefficients. They are constructed using addition, subtraction, multiplication, and non-negative integer exponents of variables. In the example provided, we have the polynomial \(-5x^3 - 6x^2 - 4x + 2\). Polynomials are essential in algebra as they form the basis of many equations and functions that we analyze.

• Each term in a polynomial is made of a coefficient (number) and a variable part, often with an exponent.
• The degree of the polynomial is determined by the highest exponent of the variable.
• Understanding polynomials helps in simplifying complex expressions and solving algebraic equations.

In algebraic equations, identifying the degree is crucial in determining if an equation is quadratic. Quadratic equations specifically have a degree of 2.

Algebraic Equations

Algebraic equations involve variables and can consist of polynomials. They are solved to find the values of the variables that satisfy the equation. The provided equation is \(-5x^3 - 6x^2 - 4x + 2 = 0\). Knowing how to rearrange and interpret these can help solve various mathematical problems.

• An equation is called quadratic if the highest exponent of the variable is 2. If it's 3, for example, as in \(-5x^3\), it would be a cubic equation.
• Understanding and identifying the type of algebraic equation is crucial for applying suitable methods to solve them.
• The goal is often to isolate the variable on one side to determine its possible values.

In the original problem, identifying the degree helps decide the correct classification of the equation.

Fraction Clearance

Fraction clearance is a technique used to remove fractions from an equation. This simplifies the process of solving algebraic equations. In the given exercise, the equation \(\frac{2}{x} - 5x^2 = 6x + 4\) involved a fraction which was cleared by multiplying all terms by \(x\).

• This technique helps convert equations with fractions into polynomial equations without fractions.
• Care must be taken to multiply each term correctly to ensure the integrity of the equation is preserved.
• By clearing fractions, complex algebraic manipulations become simpler and more straightforward.

In our problem, this technique was essential for rewriting the equation in a form where the powers of \(x\) could be easily recognized, which is necessary for determining the type of equation we have.