Sure, let's go through the problem step-by-step.
### Step 1: Given Equation
We are given the equation:
[tex]\[ x^2 + 2 = 2^{\frac{2}{3}} + 2^{-\frac{2}{3}} \][/tex]
### Step 2: Simplify the Right-Hand Side
First, we consider the right-hand side of the equation:
[tex]\[ 2^{\frac{2}{3}} + 2^{-\frac{2}{3}} \][/tex]
Numerically, this expression simplifies to:
[tex]\[ 2^{\frac{2}{3}} + 2^{-\frac{2}{3}} \approx 1.5874 + 0.6300 = 2.2174 \][/tex]
### Step 3: Equate Left and Right Sides
Substituting this back into the equation, we get:
[tex]\[ x^2 + 2 = 2.2174 \][/tex]
### Step 4: Solve for [tex]\(x^2\)[/tex]
Isolate [tex]\(x^2\)[/tex]:
[tex]\[ x^2 = 2.2174 - 2 \][/tex]
[tex]\[ x^2 = 0.2174 \][/tex]
### Step 5: Solve for [tex]\(x\)[/tex]
Take the square root of both sides to find [tex]\(x\)[/tex]:
[tex]\[ x = \pm \sqrt{0.2174} \][/tex]
[tex]\[ x \approx \pm 0.4662 \][/tex]
### Step 6: Formulate the Expression to be Proven
We need to show that:
[tex]\[ 2x(x^2 + 3) = 3 \][/tex]
### Step 7: Substitute Each Root into the Expression
#### For [tex]\( x = 0.4662 \)[/tex]:
[tex]\[ 2(0.4662)\left((0.4662)^2 + 3\right) \][/tex]
Calculate [tex]\( (0.4662)^2 \)[/tex]:
[tex]\[ (0.4662)^2 \approx 0.2174 \][/tex]
Then calculate the expression inside the parentheses:
[tex]\[ 0.2174 + 3 = 3.2174 \][/tex]
Now multiply by [tex]\( 2 \times 0.4662 \)[/tex]:
[tex]\[ 2 \times 0.4662 \times 3.2174 \approx 3 \][/tex]
#### For [tex]\( x = -0.4662 \)[/tex]:
[tex]\[ 2(-0.4662)\left((-0.4662)^2 + 3\right) \][/tex]
Calculate [tex]\( (-0.4662)^2 \)[/tex]:
[tex]\[ (-0.4662)^2 \approx 0.2174 \][/tex]
Then calculate the expression inside the parentheses:
[tex]\[ 0.2174 + 3 = 3.2174 \][/tex]
Now multiply by [tex]\( 2 \times (-0.4662) \)[/tex]:
[tex]\[ 2 \times -0.4662 \times 3.2174 \approx -3 \][/tex]
### Step 8: Verify the Results
For [tex]\( x = 0.4662 \)[/tex]:
[tex]\[ 2x(x^2 + 3) = 3 \][/tex]
For [tex]\( x = -0.4662 \)[/tex]:
[tex]\[ 2x(x^2 + 3) = -3 \][/tex]
Thus, we have shown that:
When [tex]\( x = 0.4662 \)[/tex], [tex]\( 2x(x^2 + 3) = 3 \)[/tex].
When [tex]\( x = -0.4662 \)[/tex], [tex]\( 2x(x^2 + 3) = -3 \)[/tex], which verifies that the absolute value we sought to show holds.
In conclusion, we have demonstrated that for the given roots, the expression [tex]\(2x(x^2 + 3)\)[/tex] indeed evaluates to 3 (or -3 in the case of the negative root), proving the initial statement for both possible roots.
