Solve (x-3)^2=5
The Correct Answer and Explanation is :
To solve the equation ((x – 3)^2 = 5), follow these steps:
Step 1: Remove the square by taking the square root of both sides.
Since ((x – 3)^2 = 5), we can take the square root of both sides to undo the square on the left-hand side. However, remember that when you take the square root of a number, you need to account for both the positive and negative roots.
[
\sqrt{(x – 3)^2} = \pm \sqrt{5}
]
This simplifies to:
[
x – 3 = \pm \sqrt{5}
]
Step 2: Isolate (x).
To solve for (x), we now need to add 3 to both sides of the equation to isolate (x). This gives:
[
x = 3 \pm \sqrt{5}
]
Step 3: Express the two possible solutions.
The (\pm) symbol means there are two possible solutions: one where you add (\sqrt{5}) to 3, and one where you subtract (\sqrt{5}) from 3. Thus, the two solutions are:
[
x = 3 + \sqrt{5} \quad \text{or} \quad x = 3 – \sqrt{5}
]
Step 4: Approximate the solutions.
We can approximate (\sqrt{5}). Since (\sqrt{5} \approx 2.236), we can substitute this value to get approximate numerical solutions.
- For (x = 3 + \sqrt{5}):
[
x \approx 3 + 2.236 = 5.236
]
- For (x = 3 – \sqrt{5}):
[
x \approx 3 – 2.236 = 0.764
]
Thus, the approximate solutions to the equation ((x – 3)^2 = 5) are:
[
x \approx 5.236 \quad \text{or} \quad x \approx 0.764
]
Explanation:
This problem involves solving a quadratic equation. The first step is to recognize that the square of a binomial, ((x – 3)^2), means that (x – 3) is being multiplied by itself. To isolate (x), you must undo this operation by taking the square root of both sides, and then apply the principle that squaring a number results in both a positive and negative root. Finally, by simplifying and approximating the square root of 5, you obtain the two possible solutions for (x).