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Chapter 11: Problem 5

Use the square root property to solve each equation. $$ x^{2}=18 $$

Short Answer

Expert verified
x = \pm 3\sqrt{2}

Step by step solution

01

Understand the Equation

We are given the equation \(x^2 = 18\). The goal is to solve for \(x\) using the square root property.
02

Apply the Square Root Property

The square root property states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\). Apply this to the equation \(x^2 = 18\), which gives \(x = \pm \sqrt{18}\).
03

Simplify the Square Root

Simplify \(\sqrt{18}\). The number 18 can be factored into \(2 \times 9\), and \(\sqrt{9} = 3\). So, \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\).
04

Write the Final Solution

Substitute the simplified square root back into the equation to get \(x = \pm 3\sqrt{2}\). This represents both possible solutions for \(x\).

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Key Concepts

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

Solving Quadratic Equations
Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). The equation in this exercise simplifies to \(x^2 = 18\). In such a form, solving for \(x\) is straightforward using the square root property, especially when there is no linear term (the \(b\) term is missing). To apply the square root property, take the square root of both sides of the equation:
  • Square Root Property: \(x = \pm \sqrt{a}\) when \(x^2 = a\).
Using this, we determine that \(x = \pm \sqrt{18}\). This property is essential as it provides both the positive and negative solutions due to the square's nature. Remember that the solutions are linked to the fact that squaring either positive or negative values yields the same results.
Simplifying Square Roots
Square roots can often be simplified for easier computation and better understanding. In the equation \(x = \pm \sqrt{18}\), the number 18 is the radicand. Radicands with perfect square factors can be simplified by pulling out those factors from under the square root. Here's how you can do it:
  • Break down 18 into its prime factors: \(18 = 2 \times 3^2\).
  • Recognize that \(3^2 = 9\) is a perfect square.
  • Take the square root of the perfect square out from under the radical: \(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}\).
Simplifying helps in obtaining an expression that is often neater and more relevant to calculations or further algebraic manipulation.
Factoring Numbers
Factoring is a valuable skill in simplifying roots and solving equations. To factor a number means to break it down into its basic building blocks, usually prime numbers. In our example, the task involves factoring 18 to simplify \(\sqrt{18}\):
  • Identify 18's prime factors: \(18 = 2 \times 9\) where 9 is also \(3 \times 3\).
  • So, \(18 = 2 \times 3^2\).
  • This means that the number can be rewritten to highlight its perfect square: \(3^2\).
Understanding and applying factoring can reveal hidden perfect squares within a number. This assists in simplifying square roots more effectively. Factoring thus plays a key role in various types of mathematical problems and solution strategies.

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Most popular questions from this chapter

Use the quadratic formula to solve each equation. These equations have real solutions and complex, but not real, solutions. $$ 9 x^{2}+x+2=0 $$

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