Problem 54 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (p-5)^{2}=40 $$

Step-by-Step Solution

Answer

p = 5 \pm 2 \times \sqrt{10}

Step 1: Isolate the squared term

Ensure the term with the square, \( (p-5)^2 \), is isolated on one side of the equation. In this case, it already is isolated: \( (p-5)^2 = 40 \).

Step 2: Apply the square root property

Take the square root of both sides. Remember to consider both the positive and negative square roots: \( \sqrt{(p-5)^2} = \sqrt{40} \). Therefore, \( p - 5 = \pm \sqrt{40} \).

Step 3: Simplify the radical

Simplify \( \sqrt{40} \). Notice that 40 can be factored into \( 4 \times 10 \). Hence, \( \sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10} = 2 \times \sqrt{10} \).

Step 4: Solve for p

Isolate p in both equations: \( p-5 = 2 \times \sqrt{10} \) and \( p-5 = -2 \times \sqrt{10} \). Therefore, \( p = 5 + 2 \times \sqrt{10} \) and \( p = 5 - 2 \times \sqrt{10} \).

Key Concepts

solving equations

To solve the given equation using the square root property, we need to follow specific steps. First, always ensure that the squared term is isolated on one side of the equation. This makes it easier to apply the square root property. Once isolated, we take the square root of both sides of the equation. When doing this, remember to consider both the positive and negative values of the square root. This approach will give us two possible solutions for the variable. This process can be thought of as unfolding the equation step by step, to ultimately find the values that satisfy the original equation.

simplifying radicals

Simplifying radicals is an essential step in solving equations, especially when dealing with the square root property. When simplifying a radical, the first task is to break it down into its prime factors. For instance, the radical \(\root{40}\) can be expressed as \(\root{4 \times 10}\). Next, simplify by recognizing that \(\root{4}\) can be taken out of the radical, since \(\root{4} = 2\). This results in \(\root{4 \times 10} = 2 \times \root{10}\). It’s essential to practice recognizing perfect squares inside radicals, as it makes simplification much more straightforward. Improving your skills with radicals will make solving these types of problems much easier.

isolating the variable

Once you have simplified the radicals, the next crucial step is to isolate the variable in the equation. Given our example problems, let's isolate p. Initially, we had \(\root{40}\) simplified to \(\root{4 \times 10} = 2 \times \root{10}\). After taking the square root of both sides of the equation, we consider the positive and negative values: \((p - 5) = \pm 2 \times \root{10}\). Next, we isolate p by adding 5 to both sides of each equation. This results in \p = 5 + 2 \times \root{10}\ and \p = 5 - 2 \times \root{10}\. Isolating the variable is often the final step in solving the equation, producing specific numerical values as solutions. By following these structured steps, solving the equation becomes a clear and manageable process. Learning how to isolate the variable efficiently is a core technique that applies broadly in algebra.