Problem 35 - Beginning and Intermediate Algebra

Question

Solve using the square root property. Simplify all radicals. $$ r^{2}-3=0 $$

Step-by-Step Solution

Answer

The solutions are \( r = \sqrt{3} \) and \( r = -\sqrt{3} \).

Step 1: Isolate the quadratic term

Start by isolating the term with the variable squared. To do this, add 3 to both sides of the equation: \[ r^{2} - 3 + 3 = 0 + 3 \] which simplifies to \[ r^{2} = 3 \]

Step 2: Apply the square root property

Next, apply the square root property to both sides of the equation. Remember that taking the square root of both sides requires considering both the positive and negative roots: \[ r = \pm \sqrt{3} \]

Step 3: Simplify the radicals

Simplify the expression under the radical if possible. In this case, the square root of 3 is already in its simplest form. Hence, the solutions are: \[ r = \sqrt{3} \] and \[ r = -\sqrt{3} \]

Key Concepts

solving quadratic equations

Quadratic equations are equations where the variable is raised to the power of 2. These equations have the general form \[ ax^2 + bx + c = 0 \]To solve quadratic equations, there are several methods, including:

• Factoring
• Using the quadratic formula
• Completing the square
• Square root property

.In this exercise, we use the square root property. The key steps involve isolating the squared term, taking the square root of both sides of the equation, considering both positive and negative roots, and simplifying any resulting radicals. It's important to follow these steps carefully for accurate results.

square root property

The square root property is a method to solve quadratic equations where the equation can be simplified to the form \[ x^2 = k \]Here are the steps to apply the square root property:

• Start by isolating the squared term on one side of the equation. In our example, we added 3 to both sides to get \[ r^2 = 3 \]
• Next, take the square root of both sides. Remember to consider both the positive and negative roots, resulting in \[ r = \pm\root 3 \]

The reason we consider both the positive and negative roots is because squaring either positive or negative numbers results in a positive value. This method is efficient when dealing with simple quadratic equations where the variable is only squared once.

simplifying radicals

To simplify radicals, we look for perfect square factors in the number under the radical sign. Here are some steps to simplify radicals:

• Identify if the number under the radical has any perfect square factors.
• Rewrite the radical expression using these factors, if possible.
• Calculate the square root of the perfect square factors.
• Multiply or simplify the resulting values.

For example, in our given problem, when we reach \[ r = \pm\root 3 \]we observe that 3 is a prime number with no perfect square factors, suggesting that \( \root 3 \)is already in its simplest form. Always check if the radical can be simplified further before concluding the answer.