Problem 98 - College Algebra

Question

Solve each equation in Exercises \(83-108\) by the method of your choice. $$3 x^{2}-27=0$$

Step-by-Step Solution

Answer

The solutions of the equation \(3 x^{2}-27=0\) are \(x = 3, -3 \).

Step 1: Rearrange the Equation

Rearrange the equation in order to move the constant term (-27) to the other side of the equation: \(3 x^2 = 27\)

Step 2: Divide by the Coefficient

Divide both sides of the equation by the coefficient of \( x^2 \) (which is 3) to get: \( x^2 = 9\)

Step 3: Extract the Square Root

We then take the square root of both sides. Remembering that the square root of a number gives two solutions: one positive and one negative, we get: \( x = 3, -3 \)

Key Concepts

Understanding Quadratic Equations

Quadratic equations form an integral part of algebra and they might seem daunting at first, but with a clear approach, they can be tackled easily. A standard quadratic equation is represented in the form \( ax^2 + bx + c = 0 \), where \((a, b, \ and \ c)\) are known coefficients and \(x\) represents an unknown variable. The highest degree of the variable is two, which is what makes the equation 'quadratic'.

To solve such equations, we can use several methods, including factoring, completing the square, using the quadratic formula, or graphing. However, choosing the most suitable method depends on the specific form and complexity of the equation at hand. Our sample exercise \(3x^2 - 27 = 0\) is a relatively straightforward quadratic equation, where the solution path can make use of simplification and the square root method for an efficient resolution.

Factoring Quadratics

Factoring is a method used to solve quadratic equations that can be re-written as a product of two binomials. This usually involves finding two numbers that not only multiply to give the constant term \(c\) but also add up to give the middle term \(b\) when \(a=1\). When \(a\) is not equal to one, the process can be more complex and often requires practice to master.

However, in the exercise \(3x^2 - 27 = 0\), factoring is not the most efficient method to pursue because the quadratic is already in a simplified form \(ax^2 + bx + c = 0\) where \(b=0\) and \(c=-27\). In such cases, other methods, such as the square root method, can provide a more direct route to the solutions.

The Square Root Method

The square root method is one of the simplest ways to solve quadratic equations that have the form \(x^2 = k\), where \(k\) is any positive number. For our exercise, after simplifying the original equation \(3x^2 - 27 = 0\), we obtain \(x^2 = 9\), which fits this specific form perfectly.

To apply the square root method, we take the square root of both sides of the equation, keeping in mind to consider both the positive and negative square roots. Hence, for \(x^2 = 9\), the square roots of 9 are \(+3\) and \( -3\). This gives us the two possible solutions for \(x\), thus solving our quadratic equation completely. It's essential to remember that neglecting the negative solution can lead to an incomplete set of solutions for quadratic equations.