Question

Which of the following is equal to \((3 x+2)^{2}-(x+5)(x-5) ?\) (A) \(8 x^{2}+29\) (B) \(8 x^{2}+12 x-21\) (C) \(10 x^{2}+6 x+29\) (D) \(8 x^{2}+12 x+29\)

Short answer

The equivalent expression for the given equation is (D): \( 8x^{2}+12x+29 \)

Step-by-step solution

Expand the Given Terms

Expand the squares and the multiplication terms in the given equation. So, \( (3x+2)^{2} \) will be \( 9x^{2}+12x+4 \) and \( (x+5)(x-5) \) will be \( x^{2}-25 \)

Simplify the Equation

Next, simplify the equation by subtracting \( x^{2}-25 \) from \( 9x^{2}+12x+4 \). This results in \( 8x^{2}+12x+29 \)

Compare with Answer Choices

Compare the simplified equation \( 8x^{2}+12x+29 \) with the provided answer choices. It matches the choice (D): \( 8x^{2}+12x+29 \)

Key concepts

Expanding Binomials

Understanding how to expand binomials is a foundational skill in algebra. A binomial is an algebraic expression containing two terms separated by a plus or minus sign, such as \( (a + b) \). To expand a binomial squared, such as \( (a + b)^2 \) or in our example, \( (3x + 2)^2 \), we use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \).

This is essentially distributing each term in the first binomial to each term in the second binomial. For \( (3x + 2)^2 \), we treat \(3x\) as \(a\) and \(2\) as \(b\), then apply the formula, which gives us \(9x^2 + 2(3x)(2) + 2^2\) or \(9x^2 + 12x + 4\), after simplifying.

Why is This Important?

Expanding binomials allows us to manipulate and simplify expressions so they can be compared or solved. It's also a precursor to more complex concepts like factoring polynomials and understanding the functionality of algebraic expressions within graphs.

Factoring Difference of Squares

Factoring the difference of squares is another critical tool in simplifying algebraic expressions. A difference of squares occurs when you have two squared terms separated by a subtraction sign, as in \(a^2 - b^2\). The pattern for factoring is \(a^2 - b^2 = (a - b)(a + b)\).

In the exercise, we encounter \( (x + 5)(x - 5)\). The terms \(x + 5\) and \(x - 5\) are the square roots of the first and second term, respectively, making it a classic example of the difference of squares. Using the pattern above, we can rewrite \(x^2 - 25\) directly as the product of \( (x + 5)(x - 5)\).

Practical Use

Recognizing this pattern enables quicker simplification and solving of equations, especially when working with quadratic equations. This concept is frequently used to solve equations and understand the parabolic shapes on graphs by identifying their roots.

Quadratic Equations

Quadratic equations are typically in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents the unknown variable. Solving these can be done through several methods, including factoring, using the quadratic formula, or completing the square.

In our exercise, after simplifying the given expression by expanding and factoring, we arrive at an expression similar to a quadratic equation, \(8x^2 + 12x + 29\). This equation is part of our exercise's final solution, demonstrating the application of expanding and factoring concepts explained earlier. To solve a quadratic equation like this, we would set it equal to zero and apply one of the methods for solving quadratics.

Real-World Relevance

Quadratic equations not only lay the foundation for advanced mathematics but also model various real-world phenomena, such as projectile motion or market equilibrium in economics. Mastery of this concept opens doors to understanding and solving complex problems both within mathematics and in practical scenarios.