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Chapter 9: Problem 23

Solve equation by using the square root property. Simplify all radicals. \(r^{2}-3=0\)

Short Answer

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

Step by step solution

01

- Isolate the squared term

Start by isolating the squared term on one side of the equation. Add 3 to both sides of the equation: \[ r^2 - 3 + 3 = 0 + 3 \] leading to: \[ r^2 = 3 \]
02

- Apply the square root property

Use the square root property to solve for \(r\). The square root property states that if \(x^2 = k\), then \(x = \pm \sqrt{k}\). Apply this property to the equation: \[ r^2 = 3 \] Implying \[ r = \pm \sqrt{3} \]
03

- Simplify the radicals

Simplify the radicals if necessary. In this case, \( \sqrt{3} \) is already in its simplest form. Therefore, we have two solutions for \(r\): \[ r = \sqrt{3} \] and \[ r = -\sqrt{3} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The Square Root Property
The square root property is a powerful tool for solving quadratic equations. It comes into play when your equation has been simplified to something like \(x^2 = k\). In such cases, you can solve for \(x\) by taking the square root of both sides of the equation.

In mathematical terms, if \(x^2 = k\), then \(x\ = \pm \sqrt{k}\). This means you have two potential solutions: one positive and one negative.

For example, in the original exercise, \(r^2 - 3 = 0\), once simplified, becomes \(r^2 = 3\). By applying the square root property, we get \(r = \pm \sqrt{3}\).
  • Reminder: Always remember both the positive and negative solutions!
  • This property works best when the squared term is isolated on one side of the equation.

Isolating the Squared Term
Before you can use the square root property, you have to isolate the squared term. This usually means getting rid of anything else on the same side of the equation.

For example, in the equation \(r^2 - 3 = 0\), we need to isolate \(r^2\). You can do this by adding 3 to both sides of the equation:

\[ r^2 - 3 + 3 = 0 + 3 \] which simplifies to:
\[ r^2 = 3 \]

This step is critical because the square root property works directly on the squared term. If there are other numbers or terms, you need to move them to the other side first.
  • Always perform the same operation on both sides to maintain balance.
  • This makes the equation easier to solve using radicals.

Simplifying Radicals
Simplifying radicals is the last step in our process. A radical is simplified when you cannot break it down further.

In our case with \(\sqrt{3}\), it's already simplified because 3 is a prime number. However, if you had \(\sqrt{8}\), you could simplify it by factoring it into \(\sqrt{4 \times 2}\), which breaks down into \(2\sqrt{2}\).

Here are some key tips:
  • Look for perfect squares (like 4, 9, 16) within the radical.
  • Always simplify to the smallest radical terms possible.
  • Combine like terms if you have multiple radicals.

In summary, always simplify wherever possible to get the most accurate and simplest form of your answer.

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Most popular questions from this chapter

Evaluate the expression \(\sqrt{b^{2}-4 a c}\) for the given values of \(a, b,\) and \(c .\) Simplify the radicals. See Sections \(1.3,8.1,\) and 8.2 $$a=9, b=30, c=25$$

Write each quotient in standard form. $$ \frac{-5+10 i}{6+12 i} $$

Solve each quadratic equation for complex solutions by the square root property, with \(k<0 .\) Write solutions in standard form. $$ (x+6)^{2}=-7 $$

Write each quotient in standard form. $$ \frac{40}{2+6 i} $$

Evaluate each expression for \(x=3\). $$ (x-1)^{2} $$
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