Problem 67 - Algebra 2

Question

What are the values of \(x\) that satisfy the equation \(3-27 x^{2}=0 ?\) $$ \begin{array}{ll}{\text { A. } x=\pm 3} & {\text { B. } x=\pm \frac{1}{3}} \\\ {\text { C. } x=\frac{1}{9} \text { or } x=-\frac{1}{9}} & {\text { D. } x=2 \sqrt{6} \text { or } x=-2 \sqrt{6}}\end{array} $$

Step-by-Step Solution

Answer

The values of x that satisfy the equation are x = 1/3 and x = -1/3.

Step 1: Isolate the term containing x

The first step is to isolate the term containing the variable x on one side of the equation. Start by adding 27x^2 to both sides of the equation 3 - 27x^2 = 0 to get 27x^2 = 3.

Step 2: Solve for x

Divide both sides of the equation 27x^2 = 3 by 27 to isolate x^2. This gives x^2 = 1/9.

Step 3: Take the square root of both sides

To solve for x, take the square root of both sides of the equation x^2 = 1/9, remembering that there are two solutions to a square root, one positive and one negative. This gives us x = 1/3 and x = -1/3.

Key Concepts

Quadratic Equation Solutions

Quadratic equations are mathematical statements that usually consist of a variable raised to the second power — hence 'quadratic', which comes from 'quad' meaning square. A standard quadratic equation looks like \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are known values, and \( x \) represents the variable or solution we want to find.

To approach solving these types of equations, which we call 'quadratic equation solutions', we use various methods such as factoring, completing the square, the quadratic formula, or when suitable, the square root method. Each technique has its own application depending on the structure of the quadratic equation we're given. In our example equation, \( 3 - 27x^2 = 0 \), isolating the variable and then applying the square root method provides a straightforward solution path which leads to the answer options A, B, C and D.

Isolate Variable

Isolating the variable is a critical step in solving any algebraic equation, including quadratics. The goal is to manipulate the equation to have the variable \( x \) on one side and the constants on the other. This makes it easier to solve for \( x \) without complicating the equation further.

For the equation \( 3 - 27x^2 = 0 \), we start by reversing what's keeping \( x^2 \) from being on its own. Here, we add \( 27x^2 \) to both sides to get \( 27x^2 = 3 \). Next, by dividing each side by \( 27 \), we now isolate \( x^2 \) to get the simpler equation \( x^2 = \frac{1}{9} \). This move sets us up perfectly to use the square root method to find the values of \( x \) that satisfy the initial equation.

Square Root Method

The square root method is particularly useful when we have a quadratic equation arranged in the form \( x^2 = \text{some number} \). After isolating the variable and having \( x \) squared by itself on one side, as we did with \( x^2 = \frac{1}{9} \), we proceed by taking the square root of both sides.

Here's something crucial to remember: taking the square root of a number gives us two answers - one positive and one negative. This is because squaring either a positive or negative number yields the same positive result. For instance, \( (3)^2 \) and \( (-3)^2 \) both equal \( 9 \). So, applying this to our equation, the square roots of \( \frac{1}{9} \) are \( \frac{1}{3} \) and \( -\frac{1}{3} \) - giving us our two solutions for \( x \) that satisfy the original quadratic equation.