Problem 63 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ (x+2)^{4}-81=0 $$

Step-by-Step Solution

Answer

The real solutions are \(x = 1\) and \(x = -5\).

Step 1: Isolate the Power Term

Start by moving 81 to the other side of the equation to isolate the power term. \[(x+2)^4 = 81\]

Step 2: Take the Fourth Root

Since the equation is \((x+2)^4 = 81\), take the fourth root on both sides to solve for \(x + 2\).\[x + 2 = \pm \sqrt[4]{81}\] Calculate the fourth root of 81.

Step 3: Simplify the Fourth Root

Calculate \(\sqrt[4]{81}\). You can break it down as follows:\[81 = 3^4\]Hence, \(\sqrt[4]{81} = 3\). Thus, we have:\[x + 2 = \pm 3\]

Step 4: Solve for x

Set up two equations based on \(x + 2 = 3\) and \(x + 2 = -3\) to solve for \(x\).1. When \(x + 2 = 3\), subtract 2 from both sides: \[x = 3 - 2 = 1\]2. When \(x + 2 = -3\), subtract 2 from both sides: \[x = -3 - 2 = -5\]

Step 5: Verify Solutions

Verify the solutions in the original equation.For \(x = 1\): \[(1+2)^4 - 81 = 81 - 81 = 0\] (Valid)For \(x = -5\):\[(-5+2)^4 - 81 = 81 - 81 = 0\] (Valid)Both solutions satisfy the original equation.

Key Concepts

Real Solutions

Finding real solutions means figuring out the values of the variable—let's call it \(x\)—that make the whole equation true. In our case, we start with the polynomial equation:

• \((x+2)^4 - 81 = 0\)

First, we isolate the polynomial part by moving the \(81\) to the other side:

• \((x+2)^4 = 81\)

We are interested in real solutions, which are the values that aren't imaginary numbers. Imaginary numbers are more complex and involve \(i\), which is the square root of -1. Here, since \(81\) is a nice and neat perfect fourth power, it's a great place to find real numbers as solutions.
When you solve \((x+2)^4 - 81 = 0\), you're really asking yourself, "What number, when added to 2 and raised to the fourth power, equals \(81\)?" This will help us find the real number solutions for \(x\).

Fourth Roots

Understanding fourth roots is essential when you're dealing with an equation like \((x+2)^4 = 81\).
A fourth root of a number is a value that, when raised to the fourth power, gives back that number.

• In simpler terms, if you take the fourth root of \(81\), you're searching for a number that multiplied by itself four times yields \(81\).

To compute this, consider that \(81\) is actually:

• \(3^4\)

That means,

• \(\sqrt[4]{81} = 3\)

Hence, the equation becomes:

• \(x+2 = \pm 3\)

The \(\pm\) sign (plus or minus) indicates that there are two potential roots: one positive and one negative. It shows us both possible paths for finding \(x\), leading us to two real solutions for the polynomial. The positive root corresponds to adding three while the negative root corresponds to subtracting three from \(x+2\).

Verifying Solutions

Verifying solutions is the step where we plug our found \(x\) values back into the original equation to ensure they work. This step is crucial as it confirms that we've solved the equation correctly.

• For \(x = 1\), substitute back to the original:
• \((1 + 2)^4 - 81 = 81 - 81 = 0\) (Correct!)
• For \(x = -5\), do the same:
• \((-5 + 2)^4 - 81 = 81 - 81 = 0\) (Also Correct!)

When both substitutions bring us back to zero, as the equation here requires, we can be confident in our solutions. Always verify to guard against simple arithmetic mistakes or forgotten terms—it helps maintain accuracy. This process confirms these values of \(x\) meet the condition specified by the original equation—ensuring they're indeed real solutions.