Problem 37 - Elementary Algebra

Question

For the following problems, find the products. $$ \left(x+\frac{2}{5}\right)^{2} $$

Step-by-Step Solution

Answer

Question: Expand the given binomial expression: $\left(x+\frac{2}{5}\right)^{2}$ Answer: The expanded form of the given binomial expression is: $x^{2} + \frac{4}{5}x + \frac{4}{25}$.

Step 1: Recall the binomial formula

Remember that the square of a binomial (a+b)^2 can be expanded using the formula: $$(a+b)^{2} = a^{2} + 2ab + b^{2}$$ In our case, \(a=x\) and \(b=\frac{2}{5}\).

Step 2: Apply the binomial formula

Using the formula, expand the given expression: $$\left(x+\frac{2}{5}\right)^{2} = x^{2} + 2x\left(\frac{2}{5}\right) + \left(\frac{2}{5}\right)^{2}$$

Step 3: Simplify the equation

Now, simplify the equation by performing the multiplication and squaring the terms: $$x^{2} + 2x\left(\frac{2}{5}\right) + \left(\frac{2}{5}\right)^{2} = x^{2} + \frac{4}{5}x + \frac{4}{25}$$

Step 4: Write the final result

The expanded and simplified expression for the given problem is: $$\left(x+\frac{2}{5}\right)^{2} = x^{2} + \frac{4}{5}x + \frac{4}{25}$$

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients. They are composed of terms, each being a product of a constant and a variable raised to an integer power. In the expression \(x^2 + \frac{4}{5}x + \frac{4}{25}\), we can identify it as a polynomial with three terms. Each of these terms has its own characteristics:
- \(x^2\) is the quadratic term, with its highest degree being 2.- \(\frac{4}{5}x\) is the linear term, indicating it is of the first degree.- \(\frac{4}{25}\) is the constant term, as it doesn't have a variable associated with it.
Polynomials like this one are essential in various algebraic processes, and they form a foundational aspect of algebra that is used in solving equations and modeling real-world scenarios.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They represent values and can be simplified or manipulated according to the rules of algebra. The binomial expression \(\left(x+\frac{2}{5}\right)^2\) is a perfect example of an algebraic expression.
When working with algebraic expressions:

• Understand the order of operations to simplify or expand expressions properly.
• Apply algebraic identities, such as \((a+b)^2 = a^2 + 2ab + b^2\), to expand expressions into a polynomial form.
• Simplify as far as possible to make the expression easier to work with.

Algebraic expressions are not just academic—they're tools used in engineering, physics, and economics to express and solve problems.

Quadratic Equations

Quadratic equations are equations of the second degree, meaning the highest power of the variable is 2. These equations can be expressed in the standard form \(ax^2 + bx + c = 0\).
The quadratic we discussed, \(x^2 + \frac{4}{5}x + \frac{4}{25}\), while currently an expression, can be transformed into a quadratic equation by setting it equal to zero: \(x^2 + \frac{4}{5}x + \frac{4}{25} = 0\). This setup allows us to solve for the variable \(x\) using methods such as factorization, completing the square, or using the quadratic formula:

• Factoring involves expressing the quadratic as a product of its linear factors.
• Completing the square rearranges the equation to reveal a perfect square trinomial.
• The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) directly solves for \(x\), where \(a\), \(b\), and \(c\) are coefficients from the standard form.

Quadratic equations are prevalent in physics, finance, and various optimization problems, reflecting their necessity and utility in problem-solving.