Problem 64 - A Graphical Approach to College Algebra

Question

Solve each quadratic equation by completing the square. $$3 x^{2}-3 x-1=0$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1}{2} + \frac{\sqrt{21}}{6} \) and \( x = \frac{1}{2} - \frac{\sqrt{21}}{6} \).

Step 1: Move constant to the other side

Start by moving the constant term to the right side of the equation. Subtract 1 from both sides: \[ 3x^2 - 3x = 1 \]

Step 2: Divide by the leading coefficient

Divide the entire equation by 3 to make the coefficient of \(x^2\) equal to 1.\[ x^2 - x = \frac{1}{3} \]

Step 3: Complete the square

To complete the square, take half of the coefficient of \(x\), which is -1, divide by 2 to get \(-\frac{1}{2}\), and then square it, resulting in \(\frac{1}{4}\). Add \(\frac{1}{4}\) to both sides:\[ x^2 - x + \frac{1}{4} = \frac{1}{3} + \frac{1}{4} \]. Simplifying the right side, we find a common denominator and combine:\[ x^2 - x + \frac{1}{4} = \frac{7}{12} \]

Step 4: Write left side as a square

The left side of the equation \(x^2 - x + \frac{1}{4}\) is a perfect square trinomial, which can be written as:\[ (x - \frac{1}{2})^2 = \frac{7}{12} \]

Step 5: Solve for x by taking the square root

Take the square root of both sides:\[ x - \frac{1}{2} = \pm \frac{\sqrt{7}}{\sqrt{12}} \]. Simplify the square root on the right:\[ x - \frac{1}{2} = \pm \frac{\sqrt{7}}{2\sqrt{3}} \]. Rationalize the denominator:\[ x - \frac{1}{2} = \pm \frac{\sqrt{21}}{6} \]

Step 6: Solve for x, finalize the solution

Add \(\frac{1}{2}\) to both sides to solve for \(x\):\[ x = \frac{1}{2} \pm \frac{\sqrt{21}}{6} \]. Thus, the solutions are:\[ x = \frac{1}{2} + \frac{\sqrt{21}}{6} \] and \[ x = \frac{1}{2} - \frac{\sqrt{21}}{6} \]

Key Concepts

Quadratic Equations

Understanding quadratic equations is foundational in algebra. A quadratic equation is any equation that can be rearranged into the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). This equation is called "quadratic" because "quad" means square, indicating that the highest degree of the variable \( x \) is two.

Quadratic equations can be solved by various methods, such as:

• Factoring
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Graphing
• Completing the square

The method of completing the square is especially useful as it can always be applied and helps to transform the quadratic equation into a form that is easy to solve. It involves algebraic manipulation to turn the quadratic expression into a perfect square trinomial. This topic is what we will focus more on in the next sections.

Algebraic Manipulation

Algebraic manipulation is a key skill when working with quadratic equations. It involves rearranging and simplifying expressions using algebraic techniques.

Here's how it is applied in solving the equation \(3 x^{2}-3 x-1=0\) by completing the square:

• Move the Constant: First, we isolate the quadratic and linear terms by moving the constant term to the other side. This sets the stage for completing the square.
• Normalize the Coefficient: Ensure the coefficient of \(x^2\) is one by dividing the entire equation by it. This simplifies the process of completing the square.
• Complete the Square: Add and subtract the appropriate term to make a perfect square trinomial on one side.

Through these steps, algebraic manipulation makes it possible to transform the quadratic equation into a solvable form.
In our specific example, after dividing by 3, we completed the square by adding \(\frac{1}{4}\) to both sides, hence transforming the left side into a perfect square.

Perfect Square Trinomial

Creating a perfect square trinomial is the heart of the completing the square method. A perfect square trinomial is an algebraic expression that can be expressed as the square of a binomial, \((x + p)^2\).

To make a perfect square trinomial from \(x^2 - x\):

• Take half of the coefficient of \(x\), which is \(-1\), giving \(-\frac{1}{2}\).
• Then square this result to get \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\).
• Add \(\frac{1}{4}\) to both sides of the equation to maintain the equation's balance.

This turns \(x^2 - x + \frac{1}{4} \) into \((x - \frac{1}{2})^2\), a perfect square trinomial. The equation is now in a form where the square can be easily undone by taking the square root.

In this case, the complete solution proceeds by isolating \(x\) and solving the resulting linear equation. The entire process makes quadratic equations manageable and highlights the power of algebraic manipulation in problem-solving. The final expression neatly provides two values for \(x\) that satisfy the original equation.