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Chapter 11: Problem 53

Multiply and simplify. $$(x+5)(x-5)$$

Short Answer

Expert verified
x^2 - 25

Step by step solution

01

Write the expression

Begin with the given expression: (x + 5)(x - 5)
02

Apply the difference of squares formula

Use the difference of squares formula which states that (a + b)(a - b) = a^2 - b^2 Here, a = x and b = 5. Hence, substitute: (x + 5)(x - 5) = x^2 - 5^2
03

Simplify the expression

Simplify further by calculating 5 squared: 5^2 = 25 Therefore, x^2 - 25

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebra
Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In this problem, we see algebra in action through expressions involving variables. Here, the given expression is \( (x + 5)(x - 5) \).This expression is a classic case of the 'difference of squares' in algebra. Understanding algebraic expressions and knowing how to manage them is crucial in simplifying problems effectively. When solving algebra problems, it鈥檚 essential to recognize patterns and use established formulas to make the process easier.
Multiplication
When we look at \( (x + 5)(x - 5) \), we鈥檙e dealing with multiplication of binomials. This can initially seem complex, but it becomes easier with the difference of squares formula. The difference of squares states that \[ (a + b)(a - b) = a^2 - b^2 \].Here, \( a \) is \( x \) and \( b \) is \( 5 \), transforming our expression to \( x^2 - 5^2 \). Recognizing this pattern allows us to simplify the problem through straightforward multiplication and formula application. This highlights the power of understanding and applying multiplication principles in algebra.
Simplification
The final step in the problem is simplification. Starting with \( x^2 - 5^2 \), the problem requires us to simplify further by calculating \( 5^2 \). Knowing basic squares, \( 5^2 = 25 \), the expression then simplifies to \( x^2 - 25 \).This demonstrates how simplification turns a seemingly complex problem into a simple one. Breaking down expressions step-by-step ensures clarity and accuracy, which is crucial in algebra. Understanding how to appropriately simplify expressions is a key skill in solving algebraic problems efficiently.

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Most popular questions from this chapter

Sketch the graphs of \(y=-x^{2}\) and \(y=-\frac{1}{3} x^{2}\) on the same coordinate system. How would you describe the effect the coefficient \(-\frac{1}{3}\) has on the graph of \(y=x^{2} ?\)

In Exercises \(1-64\), solve each of the given equations. If the equation is quadratic, use the factoring or square root method. If the equation has no real solutions, say so. $$\left(y-\frac{1}{2}\right)^{2}=\frac{2}{3}$$

Sketch the graph of the given equation. Find the intercepts; approximate to the nearest tenth where necessary. $$y=x^{2}+3 x+2$$

Simplify as completely as possible. (Assume \(x \geq 0 .)\) $$\frac{12}{\sqrt{5}-\sqrt{3}}$$

In Exercises \(65-74\), solve the given equation. For quadratic equations, choose either the factoring method or the square root method, whichever you think is the easier to use. $$2 x^{2}+7 x-5=3 x^{2}+9 x-4$$
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