Question

solve the equation using the square root property.

\\(5x - 3)^2 = 25\\

the solution set is \\(\\{\quad\\}\\).
(simplify your answer. type an exact answer, using radicals as needed.)

Explanation:

🆕 New Concept Discovered: Square Root Property
Using square roots to undo a perfect square.

Step 1: Apply the square root property

To undo the square on the left side of the equation, take the square root of both sides. Remember that taking the square root of a constant yields both a positive and a negative solution:

$$ (5x - 3)^2 = 25 $$

$$ 5x - 3 = \pm\sqrt{25} $$

$$ 5x - 3 = \pm 5 $$

Step 2: Set up two separate equations

This gives us two linear equations to solve:

1.

$$ 5x - 3 = 5 $$

2.

$$ 5x - 3 = -5 $$

Step 3: Solve the first equation

Solve for \( x \) when the right side is positive:

$$ 5x - 3 = 5 $$

$$ 5x = 5 + 3 $$

$$ 5x = 8 $$

$$ x = \frac{8}{5} $$

Step 4: Solve the second equation

Solve for \( x \) when the right side is negative:

$$ 5x - 3 = -5 $$

$$ 5x = -5 + 3 $$

$$ 5x = -2 $$

$$ x = -\frac{2}{5} $$

Answer:

{ \frac{8}{5}, -\frac{2}{5} }