Solving (2x-8)^2=10 Using The Square Root Property A Step-by-Step Guide

Hey guys! Let's dive into solving quadratic equations using the square root property. This is a super handy method when you've got a squared term isolated on one side of the equation. We're going to break down the steps and make it crystal clear. We are going to solve the equation: $(2x - 8)^2 = 10$.

Understanding the Square Root Property

So, what exactly is the square root property? In a nutshell, it states that if you have an equation in the form $u^2 = c$, where $u$ is an algebraic expression and $c$ is a constant, then $u$ is equal to both the positive and negative square roots of $c$. Mathematically, we write this as $u = \pm \sqrt{c}$.

The beauty of this property lies in its simplicity. When we encounter equations where a squared term is nicely isolated, we can bypass factoring or using the quadratic formula, jumping straight to finding the solutions by taking square roots. It's like finding a shortcut on a math treasure map!

Steps to Apply the Square Root Property

Let's outline the general steps for using the square root property. This will give you a roadmap for tackling these types of problems:

1. Isolate the squared term: First and foremost, you need to get the squared expression by itself on one side of the equation. This might involve adding, subtracting, multiplying, or dividing terms.
2. Take the square root of both sides: Once the squared term is isolated, take the square root of both sides of the equation. Remember to include both the positive and negative square roots!
3. Solve for the variable: After taking the square roots, you'll likely have a couple of simpler equations to solve. Isolate the variable in each equation to find your solutions.
4. Simplify, if necessary: Simplify any radical expressions or fractions to present your answers in the clearest form.
5. Check your solutions: It's always a good practice to plug your solutions back into the original equation to make sure they work. This helps catch any errors you might have made along the way.

Applying the Property to Our Example: $(2x - 8)^2 = 10$

Okay, now let's put these steps into action with our equation $(2x - 8)^2 = 10$.

• Step 1: Isolate the squared term

  Guess what? It's already done! The squared term, $(2x - 8)^2$, is nicely isolated on the left side of the equation. So, we can move straight to the next step.

• Step 2: Take the square root of both sides

  Let's take the square root of both sides of the equation. Don't forget those $\pm$ signs!

  $\sqrt{(2x - 8)^2} = \pm \sqrt{10}$

  This simplifies to:

  $2x - 8 = \pm \sqrt{10}$

• Step 3: Solve for the variable

  Now, we have two separate equations to solve:

  1. $2x - 8 = \sqrt{10}$
  2. $2x - 8 = -\sqrt{10}$

  Let's solve the first equation. Add 8 to both sides:

  $2x = 8 + \sqrt{10}$

  Now, divide both sides by 2:

  $x = \frac{8 + \sqrt{10}}{2}$

  Moving on to the second equation, add 8 to both sides:

  $2x = 8 - \sqrt{10}$

  Divide both sides by 2:

  $x = \frac{8 - \sqrt{10}}{2}$

• Step 4: Simplify, if necessary

  Our solutions are already in a pretty simplified form. We could potentially write them as:

  $x = 4 + \frac{\sqrt{10}}{2}$ and $x = 4 - \frac{\sqrt{10}}{2}$

  But the fractional form is perfectly acceptable too.

• Step 5: Check your solutions

  To be absolutely sure, you could plug these solutions back into the original equation. It's a bit tedious with the square roots, but it's a solid way to verify your work.

  So, the solutions are $x = \frac{8 + \sqrt{10}}{2}$ and $x = \frac{8 - \sqrt{10}}{2}$.

More Examples and Scenarios

To really nail this concept, let's look at a few more examples and scenarios where the square root property shines.

Example 2: A Slightly Different Setup

Consider the equation $3(x + 2)^2 = 27$.

Notice that we have a coefficient in front of the squared term. No sweat! We just need to isolate that squared term first.

• Step 1: Isolate the squared term

  Divide both sides by 3:

  $(x + 2)^2 = 9$

• Step 2: Take the square root of both sides

  $\sqrt{(x + 2)^2} = \pm \sqrt{9}$

  $x + 2 = \pm 3$

• Step 3: Solve for the variable

  We have two equations:

  1. $x + 2 = 3$
  2. $x + 2 = -3$

  Solving the first:

  $x = 1$

  Solving the second:

  $x = -5$

  So, our solutions are $x = 1$ and $x = -5$.

When to Use (and Not Use) the Square Root Property

The square root property is a fantastic tool, but it's not a one-size-fits-all solution. It's most effective when:

• You have a squared term that's isolated.
• The equation is in the form $(ax + b)^2 = c$.

It's generally not the best choice when:

• You have a quadratic equation in the standard form ($ax^2 + bx + c = 0$) where $b$ is not zero. In these cases, factoring, completing the square, or the quadratic formula are better options.
• You can't easily isolate the squared term.

Common Mistakes to Avoid

Let's highlight some common pitfalls students encounter when using the square root property so you can steer clear of them:

• Forgetting the $\pm$ sign: This is the biggest mistake! Always remember that when you take the square root of both sides, you need to consider both positive and negative roots.
• Not isolating the squared term first: You must isolate the squared term before taking square roots. Otherwise, you'll be heading down the wrong path.
• Incorrectly simplifying radicals: Make sure you know how to simplify square roots correctly. Break down the radicand (the number inside the square root) into its prime factors to look for pairs.
• Making arithmetic errors: Be careful with your arithmetic when solving for the variable after taking square roots. Double-check your work to avoid simple mistakes.

Practice Makes Perfect

The key to mastering the square root property is practice, practice, practice! Work through a variety of examples, and you'll become more confident in identifying when to use this method and how to apply it correctly.

So, to recap our solution to the original problem $(2x - 8)^2 = 10$, we found that:

$x = \frac{8 + \sqrt{10}}{2}$ and $x = \frac{8 - \sqrt{10}}{2}$

These are our solutions, guys! Keep practicing, and you'll be a square root property pro in no time!