Problem 14 - Elementary Algebra

Question

Solve each quadratic equation by extraction of roots. $$ 3 a^{2}-18=0 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(3a^2-18=0\) using the extraction of roots method. Answer: The solutions to the quadratic equation are \(a_1 = \sqrt{6}\) and \(a_2 = -\sqrt{6}\).

Step 1: Rewrite the equation in the form \(a^2=c\)

First, let's rewrite the given equation, \(3a^2-18=0\), in the form \(a^2=c\). To do this, we isolate \(a^2\) by adding 18 to both sides of the equation and then dividing by 3: $$ 3a^2 = 18 \\ a^2 = 6 $$

Step 2: Take the square root of both sides

Now that we have \(a^2=6\), we can find the value of \(a\) by taking the square root of both sides. Since we're dealing with a quadratic equation, we should expect two different solutions: one positive and one negative. So, we write: $$ a = \pm \sqrt{6} $$

Step 3: Write the final solutions

The solutions to the quadratic equation \(3a^2-18=0\) using the extraction of roots method are given by: $$ a_1 = \sqrt{6}, \quad a_2 = -\sqrt{6} $$ These are the two roots of the given quadratic equation.

Key Concepts

Extraction of Roots in Quadratic Equations

To solve a quadratic equation by the extraction of roots, we need to rewrite the equation in such a way that it becomes easy to take the square root. In the given equation, \(3a^2 - 18 = 0\), we aim to isolate \(a^2\). To do this, follow these simple steps:

• Add 18 to both sides to neutralize the constant: \(3a^2 = 18\).
• Divide both sides by 3 to simplify: \(a^2 = 6\).

Once isolated, we can extract the roots by taking the square root of both sides. This process helps you find the values of the variable that solve the equation.
When you take the square root of a number, remember that there are always two possible solutions: one positive and one negative. This is because both the square of a positive number and the square of a negative number give a positive result.

Finding Quadratic Solutions

Quadratic solutions are the values that satisfy a given quadratic equation, making it true. These solutions are often referred to as "roots" of the equation. To determine these solutions, particularly by extraction of roots, ensure that your equation is in the form \(a^2 = c\).

• Once in this form, apply the square root to both sides: \(a = \pm \sqrt{c}\).
• The \(\pm\) symbol means you have two solutions: a positive and a negative root.

For the initial equation we transformed, \(a^2 = 6\), the solutions are found by taking the square root:
\(a_1 = \sqrt{6}\) and \(a_2 = -\sqrt{6}\).
Having both solutions is important because quadratic equations typically have two different solutions unless they are perfect squares.

Understanding Square Roots

Square roots are an essential concept in mathematics, especially when working with quadratic equations. The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 6, denoted as \(\sqrt{6}\), is the number that squares to reach 6.

• From the quadratic equation \(a^2 = 6\), \(\sqrt{6}\) and \(-\sqrt{6}\) are both solutions because \((\sqrt{6})^2 = 6\) and \((-\sqrt{6})^2 = 6\).
• Square roots can appear as whole numbers, fractions, or irrational numbers like \(\sqrt{6}\).

Knowing how to work with square roots is crucial because they help solve equations and understand various properties of numbers.
In solving quadratic equations, identifying the square root provides the key to finding solutions, making this process a cornerstone of algebra.