Problem 29 - Elementary Algebra

Question

\(y^{2}=47\)

Step-by-Step Solution

Answer

y = \sqrt{47} or y = -\sqrt{47}

Step 1 - Identify the Type of Equation

This is a quadratic equation in terms of variable y, where y is raised to the power of 2.

Step 2 - Isolate the Variable

To solve for y, take the square root of both sides of the equation. This will give two possible solutions because both positive and negative values of y squared will equal 47.

Step 3 - Solve for y

Taking the square root of both sides of the equation, we get: \( y = \pm \sqrt{47} \).

Step 4 - Simplify the Expression

Since 47 is a prime number and does not have any perfect square factors, the square root of 47 cannot be simplified further. Therefore, the solutions are: \( y = \sqrt{47} \) and \( y = -\sqrt{47} \).

Key Concepts

quadratic equations

A quadratic equation is any equation that can be written in the form: \( ax^2 + bx + c = 0 \), where:

• The variable is raised to the power of 2.
• \( a, b, \) and \( c \) are constants.

In this given exercise, the equation is already in its simplest form: \( y^{2} = 47 \). Here, \( a = 1 \), \( b = 0 \), and \( c = -47 \).
To solve this quadratic equation, we follow specific steps to isolate the variable and solve for its possible values.

square root

The square root of a number is a value that, when multiplied by itself, gives the original number. For example:

• The square root of 9 is 3, because \( 3 \times 3 = 9 \).
• Similarly, the square root of 16 is 4, because \( 4 \times 4 = 16 \).

In this problem, we need to find the square root of 47.
Taking the square root of both sides of the equation \( y^{2} = 47 \) gives us \( y = \pm \sqrt{47} \).

positive and negative solutions

Quadratic equations often have two solutions. This is because both positive and negative numbers, when squared, yield the same value. For example:

• \( 3^2 = 9 \)
• \( (-3)^2 = 9 \)

Therefore, when solving \( y^{2} = 47 \), we get:

• \( y = \sqrt{47} \) (positive solution).
• \( y = -\sqrt{47} \) (negative solution).

Both solutions are valid because squaring either value will result in 47.

simplification of square roots

Simplification of square roots involves rewriting the square root expression into its simplest form.
For instance, \( \sqrt{50} \) can be simplified because 50 is 25 (a perfect square) times 2. Thus, \( \sqrt{50} = 5\sqrt{2} \).
However, not all numbers can be simplified.
In the exercise, 47 is a prime number, meaning it has no perfect square factors other than 1.
Therefore, \( \sqrt{47} \) cannot be simplified further, and the final solutions to the equation \( y^{2} = 47 \) are: \( y = \sqrt{47} \) and \( y = -\sqrt{47} \).