Problem 61 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ (3 k+1)^{2}=18 $$

Step-by-Step Solution

Answer

k = \sqrt{2} - \frac{1}{3} or k = -\sqrt{2} - \frac{1}{3}

Step 1: Isolate the squared term

The equation given is: \((3k + 1)^2 = 18\) The squared term, \((3k + 1)^2\), is already isolated.

Step 2: Apply the square root property

Take the square root of both sides of the equation to eliminate the square. \[ \sqrt{(3k + 1)^2} = \pm \sqrt{18} \] This simplifies to: \[3k + 1 = \pm \sqrt{18}\]

Step 3: Simplify the radical

Simplify \sqrt{18} to \sqrt{9 \times 2} = \sqrt{9} \sqrt{2} = 3 \sqrt{2}. Thus, the equation becomes: \[3k + 1 = \pm 3 \sqrt{2} \]so \[3k + 1 = 3 \sqrt{2} \text{or} \ 3k + 1 = -3 \sqrt{2}\]

Step 4: Solve for k

To find the value of k, solve the two equations separately: For \ 3k + 1 = 3 \sqrt{2}: \[3k = 3 \sqrt{2} - 1 \]\[ k = (3 \sqrt{2} - 1) / 3\] or \ k = \sqrt{2} - 1/3 For \ 3k + 1 = -3 \sqrt{2}: \[3k = -3 \sqrt{2} - 1 \] and solving, we get \[ k = (-3 \sqrt{2} - 1) / 3]\]k = (- \sqrt{2} - 1/3)

Key Concepts

Square Root Property

The square root property is a powerful tool for solving quadratic equations, especially those in the form of ewline (x - a)^2 = b.ewline To use the square root property, follow these steps: ewline - Isolate the squared term. - Take the square root of both sides. - Remember to include both the positive and negative roots. ewline For example, in the given exercise, we start with ewline (3k + 1)^2 = 18. ewline The term (3k + 1) is squared, and it is isolated on one side of the equation. Applying the square root property:ewline ewline ewline ewline ewline ewline ewline ewline ewline Finally, the equation becomes: 3k + 1 = ±ewline Remember to solve for both the positive and negative roots separately.

Simplifying Radicals

Simplifying radicals is crucial for solving quadratic equations accurately. When you take the square root of a number, you often get a radical, which may not always be in its simplest form. ewline For instance, in our exercise, after taking the square root of both sides of ewline (3k + 1)^2 = 18, we get ewline . This can be simplified further. ewline - Break down the number under the radical into its prime factors. - Group the factors into pairs. - Move one factor from each pair outside the radical. ewline So for 18 we can write it as . This simplifies to thus giving us 3√2. ewline By simplifying the radicals, we make the equation easier to solve. The simplified equation would then be:ewline 3k + 1 = ±3√2.

Solving for k

Once we have our equation in the form ewline 3k + 1 = ±3√2, it's time to solve for k.ewline Here's a clear process to follow: - Split the equation into two separate equations: 3k + 1 = 3√2 and 3k + 1 = -3√2. - In each equation, isolate the term with k. For 3k + 1 = 3√2, subtract 1 from both sides: ewline 3k = 3√2 - 1. Divide every term by 3: k = (3√2 - 1)/3. Similarly, solve the second equation: 3k + 1 = -3√2. Subtract 1: 3k = -3√2 - 1. Divide by 3: k = (-3√2 - 1)/3.So, we get two possible values for k: ewline k = √2 - 1/3 andk = -√2 - 1/3.It's essential to solve for k in both scenarios to find all possible solutions.