Problem 7 - Fundamentals of Mathematics

Question

Find the value of each expression. $$ y^{2}+9 y+1, \text { if } y=0. $$

Step-by-Step Solution

Answer

The value of the expression is 1 when \( y = 0 \).

Step 1: Substitute the Given Value

Since we need to evaluate the expression \( y^2 + 9y + 1 \) for \( y = 0 \), we start by substituting \( 0 \) in place of \( y \). We now have: \( 0^2 + 9(0) + 1 \).

Step 2: Simplify Each Term

Next, simplify the expression term by term. For \( 0^2 \) we have 0, and for \( 9(0) \) we also have 0. Thus, the expression simplifies to \( 0 + 0 + 1 \).

Step 3: Calculate the Result

Add up the simplified terms to get the result. Since both the \( y^2 \) term and the \( 9y \) term simplify to 0, we are left with \( 1 \). Therefore, the expression evaluates to 1.

Key Concepts

Substitution Method

The substitution method in algebra involves replacing variables with their given numerical values in an expression. This method is very helpful when you need to calculate the specific value of an algebraic expression, like in our example where we have the expression \(y^2 + 9y + 1\) and need to find its value when \(y = 0\).

Here's how the substitution method works:

• Identify the variable you need to replace and the value it is to be substituted with.
• Replace every instance of the variable in the expression with the given number.

For the expression \(y^2 + 9y + 1\), substituting \(y = 0\) results in replacing every "\(y\)" with "0". So, wherever "\(y\)" appeared originally, you now have "0". This transforms the original expression into \(0^2 + 9(0) + 1\). Now you're ready to move onto simplifying it.

Simplifying Expressions

Simplifying an expression involves performing all possible arithmetic operations to bring it to its simplest form. Once you have substituted your values in the expression, the next step is to simplify it.

Let's break down the simplification process for the expression \(0^2 + 9(0) + 1\):

• First, calculate \(0^2\), which equals 0.
• Next, multiply \(9\) by \(0\) to get 0.
• Lastly, add the remaining numbers: \(0 + 0 + 1\).

This gives us simply \(1\). Simplifying makes the expression easier to understand and use, especially when solving equations. It's crucial to always complete arithmetic operations from powers, multiplication, to addition or subtraction in order according to the math operations hierarchy known as PEMDAS/BODMAS.

Quadratic Expressions

Quadratic expressions are a type of polynomial where the highest exponent of the variable is 2. The general form of a quadratic expression is \(ax^2 + bx + c\), where "\(a\)", "\(b\)", and "\(c\)" are constants.

In our case, the expression \(y^2 + 9y + 1\) is a quadratic expression featuring the squared term \(y^2\), the linear term \(9y\), and the constant term \(1\).

• The term \(y^2\) indicates the quadratic nature because it involves the variable "y" raised to the power of two.
• The term \(9y\) is linear because it involves "y" raised to the first power.
• The term \(1\) is a constant as it doesn’t include "y".

Quadratic expressions can be evaluated by substituting values into the variables and then simplifying. These expressions often relate to quadratic equations, which are solved differently, typically involving finding the roots or solutions where the expression equals zero.