Question

2b) solve the equation ((x - 3)^2 = 25) by inverse operations.

Explanation:

Step1: Apply square root inverse

To undo the square operation, take the square root of both sides. The square root of a number has two solutions (positive and negative). So we have:
$$\sqrt{(x - 3)^2}=\pm\sqrt{25}$$
Simplifying, this becomes:
$$x - 3=\pm5$$

Step2: Solve for x (case 1: positive root)

When we take the positive square root, we have the equation \(x - 3 = 5\). To solve for \(x\), add 3 to both sides (the inverse of subtracting 3):
$$x=5 + 3$$
$$x = 8$$

Step3: Solve for x (case 2: negative root)

When we take the negative square root, we have the equation \(x - 3=-5\). Again, add 3 to both sides:
$$x=-5 + 3$$
$$x=-2$$

Answer:

The solutions are \(x = 8\) and \(x=-2\)