Problem 55 - Elementary Algebra

Question

\((x-9)^{2}=100\)

Step-by-Step Solution

Answer

x = 19 or x = -1

Step 1 - Identify the equation

The given equation is \( (x-9)^2 = 100 \).

Step 2 - Take the square root of both sides

Take the square root of both sides of the equation to eliminate the square term. This yields \( \sqrt{(x-9)^2} = \sqrt{100} \). Since we are dealing with squares, both the positive and negative roots must be considered: \( x-9 = \pm 10 \).

Step 3 - Solve the positive case

Solve the equation \( x-9 = 10 \). Add 9 to both sides to isolate \( x \): \( x = 19 \).

Step 4 - Solve the negative case

Solve the equation \( x-9 = -10 \. Add 9 to both sides to isolate \( x \): \ x = -1 \).

Step 5 - Write the final solution

The solutions to the equation \( (x-9)^2 = 100 \) are \( x = 19\) and \( x = -1 \).

Key Concepts

square root method

The square root method is a useful technique for solving quadratic equations of the form \( (x - a)^2 = b \). To use this method, first take the square root of both sides of the equation. This will eliminate the square on the left side and give you two possible equations.
Let's consider the exercise: \( (x-9)^2 = 100 \). Taking the square roots of both sides, we obtain \( \sqrt{(x-9)^2} = \sqrt{100} \), which simplifies to:

\t
• \( x-9 = 10 \)
\t
• \( x-9 = -10 \)

By splitting the equation into two parts, you'll have a path to find all potential solutions for \( x \).
Remember, whenever you take the square root, always consider both the positive and negative roots!

positive and negative roots

When solving quadratic equations, it’s crucial to remember the role of positive and negative roots.
This principle is essential because any positive number when squared gives the same result as its negative counterpart.
In our example, after taking the square root of both sides in \( (x-9)^2 = 100 \), we get two equations instead of one:

\t
• \( x-9 = 10 \)
\t
• \( x-9 = -10 \)

This step turns our quadratic into linear equations that are far easier to solve. Consider both scenarios to ensure you capture all possible solutions.
By doing this, you acknowledge both answers: \( x = 19 \) and \( x = -1 \).

isolating the variable

Isolating the variable is a key step in solving equations, especially linear ones.
This simply means rearranging the equation to get \( x \) alone on one side of the equation.
Let’s look at both conditions from the square root method:

\t
• From \( x-9 = 10 \), add 9 to both sides to find \( x \): \( x = 19 \).
\t
• From \( x-9 = -10 \), add 9 to both sides: \( x = -1 \).

This process isolates \( x \) and provides clear solutions. Keeping steps clear and logical will help prevent errors along the way.
Always perform the same operation on both sides of the equation to maintain balance.

quadratic solutions

Quadratic solutions can often have more than one answer as they involve squared variables, which inherently account for both positive and negative values.
In the given problem \( (x-9)^2 = 100 \), we use the square root method to find multiple solutions. By solving for \( x \), we perform these essential steps:

\t
• Take the square root of both sides.
\t
• Consider the positive and negative roots.
\t
• Isolate the variable to find the values of \( x \).

Finally, the solutions are \( x = 19 \) and \( x = -1 \).
This example highlights the nature of quadratic equations and the importance of following each step carefully to find all possible solutions.