Given $(x-1)^2=50$, select the values of $x$. Choose two correct answers.
The given equation is \((x-1)^2 = 50\). This is a quadratic equation where we need to isolate the variable \(x\). We observe that the left side is a perfect square. To solve for \(x\), we can take the square root of both sides of the equation, remembering that the square root of a positive number yields both a positive and a negative result.
Taking the square root of both sides of the equation \((x-1)^2 = 50\), we get:
\[x - 1 = \pm \sqrt{50}\]
Next, we simplify the square root of 50. Since \(50 = 25 \times 2\), we have:
\[\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}\]
Substituting this back into our equation, we obtain:
\[x - 1 = \pm 5\sqrt{2}\]
Now we solve for \(x\) by adding 1 to both sides of the equation:
\[x = 1 \pm 5\sqrt{2}\]
This gives us two distinct solutions:
Comparing these results with the provided options:
We can clearly see that options a and d are the correct solutions.
Quadratic equations like these are common when calculating distances or areas in geometry. For example, if you know the area of a square patch of land is 50 square meters, you might need to find the side length by solving for it. If the side length is represented by \((x-1)\), then \((x-1)^2 = 50\) helps determine the exact dimensions. This technique of solving for \(x\) by isolating the squared term and taking the square root is a fundamental skill in algebra used in physics, engineering, and everyday construction tasks.