Alright, let's solve the equation step-by-step!
Step 1: Simplify the Expression Inside the Equation
The given equation is:
[tex]\[ x^2 = (-2 \sqrt{6})^2 + 5^2 \][/tex]
First, let's deal with each term separately. We need to calculate [tex]\((-2 \sqrt{6})^2\)[/tex] and [tex]\(5^2\)[/tex].
Step 2: Compute [tex]\((-2 \sqrt{6})^2\)[/tex]
When you square [tex]\(-2 \sqrt{6}\)[/tex]:
[tex]\[ (-2 \sqrt{6})^2 = (-2)^2 \cdot (\sqrt{6})^2 \][/tex]
[tex]\[ = 4 \cdot 6 \][/tex]
[tex]\[ = 24 \][/tex]
So, [tex]\((-2 \sqrt{6})^2\)[/tex] simplifies to 24.
Step 3: Compute [tex]\(5^2\)[/tex]
Next, calculate:
[tex]\[ 5^2 = 25 \][/tex]
Step 4: Sum the Results
Now, add the results from steps 2 and 3:
[tex]\[ x^2 = 24 + 25 \][/tex]
[tex]\[ x^2 = 49 \][/tex]
Step 5: Solve for [tex]\(x\)[/tex]
To find [tex]\(x\)[/tex], we take the square root of both sides of the equation:
[tex]\[ x = \sqrt{49} \][/tex]
[tex]\[ x = 7 \][/tex]
Therefore, the step-by-step solution to the equation [tex]\(x^2 = (-2 \sqrt{6})^2 + 5^2\)[/tex] yields:
[tex]\[ x = 7 \][/tex]
In summary:
- [tex]\((-2 \sqrt{6})^2\)[/tex] evaluates to 24.
- [tex]\(5^2\)[/tex] evaluates to 25.
- The sum of these values is 49, meaning [tex]\( x^2 = 49 \)[/tex].
- Taking the square root of 49, we find [tex]\( x = 7 \)[/tex].
