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A quadratic equation is any equation that can be written in the form: \(ax^2 + bx + c = 0\). To solve these equations, you need to find the values of \(x\) that make the equation true. The given problem \(p^2 - 50 = 0\) is a simplified quadratic equation where \(a=1\), \(b=0\), and \(c=-50\). Follow these steps to solve such equations:

• Step 1: Isolate the squared term. Add or subtract terms to get the squared term alone on one side of the equation.
• Step 2: Take the square root of both sides. Remember, taking the square root will give you two solutions: one positive and one negative.
• Step 3: Simplify the square root. Break down the square root into simpler terms, if possible.

Square roots can be a tricky concept, but let's break it down. When you see a square root symbol (\(\sqrt{}\)), you are looking for a number that, when multiplied by itself, gives you the number under the root. For example, \(\sqrt{49} = 7\) because \(7 \times 7 = 49\).

In the case of the given problem \(\sqrt{50}\), this means you are searching for a number that multiplies by itself to result in 50. However, \(50\) is not a perfect square. In these instances, you can often simplify the square root by breaking it down into the product of known squares. For example, \(\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5 \sqrt{2}\). So, always check if the number inside the root can be broken down into squares to make your life easier.

Simplifying radicals means reducing the expression under the radical sign to its simplest form. This often involves finding the prime factors of the number and seeing if any of them are perfect squares.

Let's look at the square root of 50 again: \(\sqrt{50}\).

• Start by expressing 50 as a product of prime factors: \(50 = 25 \times 2\).
• Since 25 is a perfect square, you can take its square root: \(\sqrt{25} = 5\).
• Thus, \(\sqrt{50} = 5 \sqrt{2}\).

So, the expression \(5\sqrt{2}\) is the simplified form of \(\sqrt{50}\). This helps in reducing complex expressions and makes solving equations more straightforward. Remember, always look for perfect squares within the radical to simplify it effectively.