Problem 114 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Solve: \(4(x-5)^{2}+9=53\)

Step-by-Step Solution

Answer

The solutions are \(x = 5 + \sqrt{11}\) and \(x = 5 - \sqrt{11}\).

Step 1: Isolate the quadratic expression

Subtract 9 from both sides of the equation to isolate the quadratic expression. \[4(x-5)^2 + 9 - 9 = 53 - 9\] Simplify the equation: \[4(x-5)^2 = 44\]

Step 2: Divide both sides by 4

To further isolate the squared term, divide both sides of the equation by 4: \[\frac{4(x-5)^2}{4} = \frac{44}{4} \] Simplify the equation: \[(x-5)^2 = 11\]

Step 3: Take the square root of both sides

Apply the square root to both sides to solve for \(x-5\) \[\sqrt{(x-5)^2} = \sqrt{11}\] Consider both the positive and negative square roots: \[x-5 = \pm \sqrt{11}\]

Step 4: Solve for x

Add 5 to both sides to solve for \(x\): \[x = 5 \pm \sqrt{11}\] So, the two solutions are: \[x = 5 + \sqrt{11}\] and \[x = 5 - \sqrt{11}\]

Key Concepts

quadratic expression

A quadratic expression is any mathematical expression where the highest degree of the variable is two. In simpler terms, it includes terms like \(x^2\) or \(y^2\). Quadratic expressions can appear in various forms, such as \(ax^2 + bx + c\). In our problem, we have \(4(x-5)^2\). This expression involves squaring the binomial \(x-5\) and then multiplying by 4. Recognizing and understanding the structure of quadratic expressions is crucial. It helps us know which methods to apply when solving equations containing them.

isolating variables

Isolating the variable is a key step in solving equations. In our problem, the ultimate goal is to solve for \(x\). To do this, we need to perform a series of steps that isolate \(x\).

• First, we subtract 9 from both sides: \[4(x-5)^2 + 9 - 9 = 53 - 9\]
• Simplify to get: \[4(x-5)^2 = 44\]
• Next, to further isolate \(x-5\), we divide both sides by 4: \[\frac{4(x-5)^2}{4} = \frac{44}{4}\]
• Which simplifies to: \[(x-5)^2 = 11\]

Isolating the variable simplifies the equation step by step, making it easier to solve.

square root method

The square root method involves taking the square root of both sides of an equation to solve for the variable. In our problem, after isolating the quadratic term \((x-5)^2 = 11\), we apply the square root to both sides. This looks like: \[\root{(x-5)^2} = \root{11}\] Remember that taking the square root gives us both positive and negative roots. So, we get: \[x-5 = \pm \sqrt{11}\] Understanding and using the square root method effectively helps break down more complex quadratic equations into simpler forms that are easier to handle.

solving for x

After applying the square root method, the next step is to solve for \(x\). With the equation: \(x-5 = \pm \sqrt{11}\), we solve for \(x\) by adding 5 to both sides. This looks like: \[x = 5 \pm \sqrt{11}\] This step gives us the final solutions for \(x\): \[x = 5 + \sqrt{11}\] and \[x = 5 - \sqrt{11}\] Solving for \(x\) in this way ensures we find all possible solutions to the quadratic equation. In conclusion, understanding each of these steps breaks down the problem, making even complex-looking equations much simpler to handle.