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Quadratic equations are equations of the form \[ ax^2 + bx + c = 0 \]. These equations can be solved using various methods such as factoring, completing the square, the quadratic formula, and the square root property. The square root property is especially useful when the equation is already expressed in a form that isolates the squared term.To use the square root property, we need to isolate the squared term on one side of the equation.Then we can take the square root of both sides. Always remember that taking the square root of both sides introduces two potential solutions: one positive and one negative. This is because both positive and negative numbers when squared will yield the same positive value. Hence, the solutions will be in the form: \[ x = \pm \sqrt{k} \].We start by isolating the squared term and then apply the square root property to find the solutions.

Algebraic manipulation involves rearranging equations to isolate the desired variable.This process follows various rules of algebra such as the distributive property, combining like terms, and performing the same operation on both sides of an equation.In the given exercise, we start with the equation: \[ p^2 - 50 = 0 \].The goal is to isolate \[ p^2 \] by adding 50 to both sides of the equation: \[ p^2 - 50 + 50 = 0 + 50 \].This simplifies to: \[ p^2 = 50 \].By understanding and correctly applying algebraic principles, we can transform the equation step by step until we isolate the variable, making it easier to solve.

Simplifying square roots involves breaking down the number inside the square root into its prime factors. Then, group the factors into pairs to simplify. Consider the square root we encounter in the exercise: \[ \sqrt{50} \].We can express 50 as the product of its prime factors: \[ 50 = 25 \times 2 \]Next, we know that the square root of a product is the product of the square roots: \[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} \].We recognize that \[ 25 \] is a perfect square, and \[ \sqrt{25} = 5 \],thus simplifying to: \[ \sqrt{50} = 5 \sqrt{2} \].Finally, the solutions to the equation are \[ p = \pm 5 \sqrt{2} \].Breaking down and simplifying square roots is an essential skill in solving quadratic equations.