Solving Sqrt(x+12) = X: Step-by-Step Solution

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Hey guys! Today, we're diving into a fun little math problem: figuring out the solution to the equation sqrt(x+12) = x. This type of problem involves a square root, which might seem a bit intimidating at first, but don't worry, we'll break it down step-by-step so it's super easy to understand. So, grab your pencils, and let's get started!

Understanding the Problem

Before we jump into the solution, let's make sure we understand what the problem is asking. We need to find the value(s) of 'x' that make the equation sqrt(x+12) = x true. The key here is to remember that the square root of a number is always non-negative (i.e., zero or positive). This is crucial because it will help us eliminate any extraneous solutions later on.

Why is Understanding the Square Root Important?

When dealing with square roots, we need to be extra careful because squaring both sides of an equation can sometimes introduce solutions that don't actually work in the original equation. These are called extraneous solutions. Think of it like this: if we have an equation A = B, then A² = B² is true, but the reverse isn't always true. For example, if A = -2 and B = 2, then A² = 4 and B² = 4, so A² = B², but A ≠ B. Therefore, it's super important to check our answers at the end to make sure they're the real deal!

Initial Thoughts

Looking at the equation sqrt(x+12) = x, we can immediately see a couple of things. First, the expression inside the square root (x+12) must be non-negative, which means x+12 ≥ 0, or x ≥ -12. Second, since the square root is always non-negative, x itself must also be non-negative (x ≥ 0). This gives us a starting point for where to look for our solutions. We know x must be greater than or equal to zero, which will help later when verifying solutions.

Step 1: Squaring Both Sides

The first step in solving this equation is to get rid of the square root. We can do this by squaring both sides of the equation. This gives us:

(sqrt(x+12))² = x²

This simplifies to:

x + 12 = x²

Why Square Both Sides?

Squaring both sides is a common technique for solving equations involving square roots. It's like undoing the square root operation. Just remember, we need to be careful about extraneous solutions, which is why the next steps are so important.

Step 2: Rearranging the Equation

Now we have a quadratic equation, which is an equation of the form ax² + bx + c = 0. To solve it, we need to rearrange our equation into this standard form. We can do this by subtracting x and 12 from both sides:

x² - x - 12 = 0

Spotting the Quadratic Form

Recognizing the quadratic form is key because we have a bunch of different methods to solve these types of equations. We could use factoring, the quadratic formula, or even completing the square. For this particular equation, factoring looks like the easiest route.

Step 3: Factoring the Quadratic

We need to factor the quadratic expression x² - x - 12. This means we need to find two numbers that multiply to -12 and add up to -1. Those numbers are -4 and 3. So, we can factor the quadratic as:

(x - 4)(x + 3) = 0

Factoring Explained

Factoring is like reverse multiplication. We're trying to find two expressions that, when multiplied together, give us our original quadratic expression. This step turns the problem into something much easier to solve.

Step 4: Finding Potential Solutions

Now that we have the factored form, we can use the zero-product property, which says that if the product of two factors is zero, then at least one of the factors must be zero. So, either (x - 4) = 0 or (x + 3) = 0. This gives us two potential solutions:

  • x - 4 = 0 => x = 4
  • x + 3 = 0 => x = -3

The Zero-Product Property

The zero-product property is a powerful tool for solving factored equations. It allows us to break down a single equation into multiple simpler equations, each of which we can solve individually.

Step 5: Checking for Extraneous Solutions

This is the most critical step! We need to plug our potential solutions back into the original equation sqrt(x+12) = x to see if they actually work. Remember, squaring both sides can introduce extraneous solutions, so we need to be sure.

Checking x = 4

Let's plug x = 4 into the original equation:

sqrt(4 + 12) = 4

sqrt(16) = 4

4 = 4

This is true, so x = 4 is a valid solution!

Checking x = -3

Now let's try x = -3:

sqrt(-3 + 12) = -3

sqrt(9) = -3

3 = -3

This is not true! Remember, the square root of a number is always non-negative. So, x = -3 is an extraneous solution, and we discard it.

Why Checking is Crucial

Checking for extraneous solutions is a non-negotiable step when solving equations involving square roots (or any radical equations). Without this check, we might end up with answers that don't actually satisfy the original equation.

Conclusion: The Solution

After going through all the steps, we've found that the only valid solution to the equation sqrt(x+12) = x is x = 4. The potential solution x = -3 turned out to be an extraneous solution.

Key Takeaways

  • Solving equations with square roots involves isolating the square root and squaring both sides.
  • Squaring both sides can introduce extraneous solutions, so always check your answers in the original equation.
  • Factoring quadratic equations is a useful skill for solving many types of problems.
  • The zero-product property helps us find solutions once we have factored an equation.

So there you have it! We've successfully solved the equation sqrt(x+12) = x. Remember to always double-check your work and watch out for those pesky extraneous solutions. Keep practicing, and you'll become a pro at solving these types of problems in no time! Happy math-ing, guys!