Problem 22 - Algebra and Trigonometry

Question

Solve the equation by completing the square. \(3 x^{2}-6 x-1=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 1 + \frac{2\sqrt{3}}{3}\) and \(x = 1 - \frac{2\sqrt{3}}{3}\).

Step 1: Move the Constant Term to the Other Side

Start by moving the constant term to the right side of the equation. The equation becomes: \[ 3x^2 - 6x = 1 \]

Step 2: Divide by the Coefficient of \(x^2\)

Divide the entire equation by 3, the coefficient of \(x^2\):\[ x^2 - 2x = \frac{1}{3} \]

Step 3: Complete the Square

To complete the square, take half of the coefficient of \(x\) (which is -2), square it, and add it to both sides. Half of -2 is -1, and \((-1)^2 = 1\). So add 1 to both sides:\[ x^2 - 2x + 1 = \frac{1}{3} + 1 \] \[ x^2 - 2x + 1 = \frac{4}{3} \]

Step 4: Write the Left Side as a Squared Binomial

The left side is now a perfect square trinomial, which can be written as:\[ (x - 1)^2 = \frac{4}{3} \]

Step 5: Take the Square Root of Both Sides

Take the square root of both sides to solve for \(x\):\[ x - 1 = \pm \sqrt{\frac{4}{3}} \]Simplify \[ x - 1 = \pm \frac{2}{\sqrt{3}} \] Rationalize the denominator:\[ x - 1 = \pm \frac{2\sqrt{3}}{3} \]

Step 6: Solve for \(x\)

Add 1 to both sides to solve for \(x\):\[ x = 1 \pm \frac{2\sqrt{3}}{3} \]

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree two. These are common in algebra and usually take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. Solving quadratic equations can be done using different methods like factoring, using the quadratic formula, or completing the square. In the exercise, the focus is on completing the square, a method that rearranges the equation into a form where one side is a perfect square trinomial.Understanding quadratic equations is crucial because they appear in various mathematical contexts, such as geometry and physics. They describe parabolas when graphed, which can model real-world situations like projectile motion or economics.By exploring various solving methods, you deepen your understanding of these equations and become more flexible in handling algebraic problems. While factoring is fast when applicable, completing the square, as we did in this exercise, is a powerful tool, especially when equations don't factor easily.

Rationalize the Denominator

Rationalizing the denominator simplifies fractions that contain square roots or other irrational numbers in the denominator. It's a common technique to ensure expressions are easier to read and work with. Let's break it down:

• Start with a fraction like \( \frac{2}{\sqrt{3}} \).
• Multiply both the numerator and the denominator by the square root present in the denominator, in this case, \( \sqrt{3} \).
• This results in \( \frac{2\sqrt{3}}{3} \), where the denominator is now a rational number (3).

Rationalizing is important in problems involving square roots, as it helps in achieving neat and simplified results, which are often required in final answers. This technique aligns with many mathematical conventions and aids in easier computation in other mathematical operations.

Perfect Square Trinomial

A perfect square trinomial is a trinomial that can be expressed as the square of a binomial. The general form is \( (x + d)^2 = x^2 + 2dx + d^2 \). Identifying and creating perfect square trinomials is essential when solving equations by completing the square.In the given exercise, after moving terms around and manipulating the equation, we encountered \( x^2 - 2x + 1 \), a perfect square trinomial. This expression can be written as \( (x - 1)^2 \). The conversion to a perfect square trinomial simplifies the solving process, allowing us to solve the equation by taking the square root of both sides.The concept is pervasive in mathematics as it simplifies complex expressions and aids in deriving straightforward solutions. Mastering the art of completing the square with perfect square trinomials is not only useful in algebra but also in calculus and beyond.