Certainly! Let's find an equation that is equivalent to [tex]\( x^2 - 10x = 14 \)[/tex].
### Step-by-Step Solution:
1. Starting Equation:
[tex]\[ x^2 - 10x = 14 \][/tex]
2. Rearrange the Equation:
To make it easier to solve by completing the square, first rearrange the equation to bring all terms to one side:
[tex]\[ x^2 - 10x - 14 = 0 \][/tex]
3. Complete the Square:
To complete the square, we need to form a perfect square trinomial from [tex]\( x^2 - 10x \)[/tex].
- Take the coefficient of [tex]\( x \)[/tex], which is [tex]\(-10\)[/tex], halve it to get [tex]\(-5\)[/tex], and then square it:
[tex]\[ \left(\frac{-10}{2}\right)^2 = (-5)^2 = 25 \][/tex]
- Rewrite the equation by adding and subtracting this squared value inside the equation:
[tex]\[ x^2 - 10x + 25 - 25 - 14 = 0 \][/tex]
4. Forming the Perfect Square:
The expression [tex]\( x^2 - 10x + 25 \)[/tex] is a perfect square:
[tex]\[ (x - 5)^2 - 25 - 14 = 0 \][/tex]
Simplify this:
[tex]\[ (x - 5)^2 - 39 = 0 \][/tex]
5. Isolate the Perfect Square:
Finally, isolate [tex]\( (x - 5)^2 \)[/tex] on one side of the equation:
[tex]\[ (x - 5)^2 = 39 \][/tex]
### Conclusion:
The equivalent equation to the given equation [tex]\( x^2 - 10x = 14 \)[/tex] is:
[tex]\[ (x - 5)^2 = 39 \][/tex]
Therefore, the correct answer is:
[tex]\[ (x - 5)^2 = 39 \][/tex]
