To determine which equation has the same solutions as the given equation [tex]\(2x^2 - 12x - 50 = 0\)[/tex], we need to follow these detailed steps:
### Step 1: Solve the Given Equation
The given quadratic equation is:
[tex]\[2x^2 - 12x - 50 = 0\][/tex]
First, divide the whole equation by 2 to simplify it:
[tex]\[ x^2 - 6x - 25 = 0 \][/tex]
Next, use the quadratic formula to find the solutions of the simplified equation:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 1\)[/tex], [tex]\(b = -6\)[/tex], and [tex]\(c = -25\)[/tex].
Calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac = (-6)^2 - 4(1)(-25) = 36 + 100 = 136 \][/tex]
Thus, the solutions are:
[tex]\[ x = \frac{6 \pm \sqrt{136}}{2} = \frac{6 \pm 2\sqrt{34}}{2} = 3 \pm \sqrt{34} \][/tex]
Therefore, the solutions to the given equation [tex]\(2x^2 - 12x - 50 = 0\)[/tex] are:
[tex]\[ x_1 = 3 + \sqrt{34} \][/tex]
[tex]\[ x_2 = 3 - \sqrt{34} \][/tex]
### Step 2: Verify Each Choice
Now, we need to check which of the given equations has the same solutions [tex]\(x_1 = 3 + \sqrt{34}\)[/tex] and [tex]\(x_2 = 3 - \sqrt{34}\)[/tex].
#### Choice A: [tex]\((x - 6)^2 = 19\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ (x - 6)^2 = 19 \][/tex]
[tex]\[ x - 6 = \pm \sqrt{19} \][/tex]
[tex]\[ x = 6 \pm \sqrt{19} \][/tex]
These solutions are:
[tex]\[ x_1 = 6 + \sqrt{19} \][/tex]
[tex]\[ x_2 = 6 - \sqrt{19} \][/tex]
These do not match [tex]\( x_1 = 3 + \sqrt{34} \)[/tex] and [tex]\( x_2 = 3 - \sqrt{34} \)[/tex].
#### Choice B: [tex]\((x - 3)^2 = 16\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ (x - 3)^2 = 16 \][/tex]
[tex]\[ x - 3 = \pm 4 \][/tex]
[tex]\[ x = 3 \pm 4 \][/tex]
These solutions are:
[tex]\[ x_1 = 3 + 4 = 7 \][/tex]
[tex]\[ x_2 = 3 - 4 = -1 \][/tex]
These do not match [tex]\( x_1 = 3 + \sqrt{34} \)[/tex] and [tex]\( x_2 = 3 - \sqrt{34} \)[/tex].
#### Choice C: [tex]\((x - 6)^2 = 31\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ (x - 6)^2 = 31 \][/tex]
[tex]\[ x - 6 = \pm \sqrt{31} \][/tex]
[tex]\[ x = 6 \pm \sqrt{31} \][/tex]
These solutions are:
[tex]\[ x_1 = 6 + \sqrt{31} \][/tex]
[tex]\[ x_2 = 6 - \sqrt{31} \][/tex]
These do not match [tex]\( x_1 = 3 + \sqrt{34} \)[/tex] and [tex]\( x_2 = 3 - \sqrt{34} \)[/tex].
#### Choice D: [tex]\((x - 3)^2 = 34\)[/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[ (x - 3)^2 = 34 \][/tex]
[tex]\[ x - 3 = \pm \sqrt{34} \][/tex]
[tex]\[ x = 3 \pm \sqrt{34} \][/tex]
These solutions are:
[tex]\[ x_1 = 3 + \sqrt{34} \][/tex]
[tex]\[ x_2 = 3 - \sqrt{34} \][/tex]
### Conclusion
The solutions for choice D match the solutions for the given equation.
Thus, the correct answer is:
[tex]\[ \boxed{(x - 3)^2 = 34} \][/tex]
or
[tex]\[ \text{Choice D} \][/tex]
