Question

If \(x=5,\) why does \(x^{2}+x=30 ?\)

Short answer

When \(x=5\), \(x^2 + x = 25 + 5 = 30\).

Step-by-step solution

Substitute the value of x

Given that \(x=5\), substitute \(5\) in place of \(x\) in the expression \(x^2 + x\). This gives us the following equation: \(5^2 + 5\).

Calculate the square of x

Calculate \(5^2\). Here, \(5^2\) means \(5\) multiplied by \(5\): \(5 \times 5 = 25\).

Add x to the result

Add \(5\) to the result obtained from squaring \(5\). So, we have \(25 + 5\).

Simplify the expression

Simplify \(25 + 5\) to get \(30\).

Conclusion

Since substituting \(x = 5\) into the expression \(x^2 + x\) results in \(30\), this verifies that \(x^2 + x = 30\) when \(x = 5\).

Key concepts

substitution method

The substitution method is a valuable technique in algebra used to solve equations by replacing variables with their known values. It simplifies expressions and makes calculations easier. When you substitute a value into an equation, you effectively transform the problem into a simpler arithmetic task.

Example: Given the expression \( x^2 + x \) and knowing that \( x = 5 \), you substitute 5 in place of \( x \) to get \( 5^2 + 5 \). This method allows you to convert an algebraic problem into a numerical one.

By performing substitution correctly, you ensure your solutions are accurate and make complex problems more manageable. A crucial part of this method involves careful substitution and then following through with arithmetic calculations. This streamlines the problem-solving process and helps in checking if an expression holds true for a given value.

simplifying expressions

Simplifying expressions involves combining like terms and carrying out arithmetic operations to make the expression as concise and easy to work with as possible. This is a fundamental skill in algebra that helps in solving equations efficiently.

Starting with an expression like \( 5^2 + 5 \), you simplify by first calculating \( 5^2 \), which is \( 25 \). You then add the remaining term, \( 5 \), to get \( 25 + 5 = 30 \).

Steps for Simplifying:

• Carry out exponentiation or multiplication first (following order of operations)
• Combine any like terms that are present
• Perform any addition or subtraction

Simplifying expressions reduces complexity and brings you closer to the final solution. This practice is especially useful when dealing with longer and more complicated algebraic expressions. By breaking down the expression into smaller, manageable parts, algebra becomes much more straightforward.

basic algebra

Basic algebra forms the foundation for all higher mathematics and involves understanding variables, constants, and basic operations. It's crucial to grasp the fundamental concepts to solve equations and simplify expressions effectively.

Core concepts include:

• Variables: Symbols like \( x \) that represent unknown values
• Constants: Known values like 5
• Operations: Addition, subtraction, multiplication, and division

For instance, in the problem \( x^2 + x \), you identify \( x \) as a variable and 2 as an exponent indicating multiplication. Substitution is used here to replace \( x \) with a constant value, and simplification follows basic arithmetic rules.

Mastering these fundamentals enables you to handle more complex algebraic problems as you advance in math. Regular practice with simpler problems ensures these techniques become second nature, allowing you to solve equations and manipulate expressions with confidence.