Problem 52 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (x+3)^{2}=11 $$

Step-by-Step Solution

Answer

x = -3 \pm \sqrt{11}

Step 1 - Isolate the Squared Term

The squared term \( (x+3)^{2} \) is already isolated on the left-hand side of the equation \( (x+3)^{2}=11 \).

Step 2 - Apply the Square Root Property

Take the square root of both sides of the equation to eliminate the square. Remember to include both the positive and negative square roots: \[ \sqrt{(x+3)^{2}} = \pm \sqrt{11} \Rightarrow x+3 = \pm \sqrt{11} \]

Step 3 - Solve for x

To isolate x, subtract 3 from both terms on the right-hand side: \[ x = -3 \pm \sqrt{11} \]

Step 4 - Simplify the Radicals

The \sqrt{11} is already in its simplest form, so your final simplified solutions for x are: \[ x = -3 + \sqrt{11} \] \[ x = -3 - \sqrt{11} \]

Key Concepts

square root property

The square root property is an essential tool for solving quadratic equations. It states that if we have an equation in the form \(x^2 = k\), we can solve it by taking the square root of both sides. This gives us two potential solutions, since the square root of a number can be positive or negative. For example, \(x^2 = 11\) leads to \(x = \pm\sqrt{11}\). In our original exercise, we apply this property to \( (x+3)^2 = 11 \) to find \( x+3 = \pm\sqrt{11} \). Always remember to include both positive and negative roots when applying the square root property.

isolating the variable

Isolating the variable is a crucial step in solving equations. This means getting the variable by itself on one side of the equation. For our exercise, we start with \( (x+3)^2 = 11 \). Luckily, the squared term \( (x+3)^2 \) is already isolated. After taking the square root of both sides, \( x+3 = \pm\sqrt{11} \), we further isolate x by subtracting 3 from both sides. This leaves us with \( x = -3 \pm\sqrt{11} \). Properly isolating the variable ensures we correctly solve for the unknown.

simplifying radicals

Simplifying radicals means expressing the radical in its simplest form. In our problem, \(\sqrt{11}\) is already simplified, as 11 is a prime number and has no square factors other than 1. Take, for example, \( \sqrt{12} \): we can break it down into \(\sqrt{4 \times 3} \), which simplifies further to \ 2\sqrt{3}\. Simplifying radicals makes them easier to work with in later steps and ensures your final answer is in its most reduced form.

positive and negative roots

When solving equations using the square root property, we must always consider both positive and negative roots. For instance, in the problem \( (x+3)^2 = 11 \), taking the square root of both sides gives us \( x+3 = \pm\sqrt{11} \). This results in two solutions: \ x+3 = \sqrt{11} \ and \ x+3 = -\sqrt{11} \. Solving each, we find \ x = -3 + \sqrt{11} \ and \ x = -3 - \sqrt{11}\. Including both roots is essential for capturing all possible solutions, as neglecting one might lead to an incomplete answer.