Problem 35 - Algebra 2

Question

Solve each equation by completing the square. \(9 x^{2}-6 x-4=0\)

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1+\sqrt{5}}{3}\) and \(x = \frac{1-\sqrt{5}}{3}\).

Step 1: Start with the Quadratic Equation

The given quadratic equation is \( 9x^2 - 6x - 4 = 0 \). Our goal is to solve this equation by completing the square.

Step 2: Move the Constant Term

Rearrange the equation to move the constant to the other side: \[ 9x^2 - 6x = 4 \]

Step 3: Make the Quadratic Coefficient 1

To complete the square, the coefficient of \(x^2\) should be 1. Divide the entire equation by 9: \[ x^2 - \frac{2}{3}x = \frac{4}{9} \]

Step 4: Complete the Square

To complete the square, take half of the coefficient of \(x\) (\(-\frac{2}{3}\)), square it, and add it to both sides. Half of \(-\frac{2}{3}\) is \(-\frac{1}{3}\), and squaring it gives \(\frac{1}{9}\). Add \(\frac{1}{9}\) to both sides: \[ x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{4}{9} + \frac{1}{9} \] \[ x^2 - \frac{2}{3}x + \frac{1}{9} = \frac{5}{9} \]

Step 5: Express as a Perfect Square

The left side is now a perfect square: \[ \left(x - \frac{1}{3}\right)^2 = \frac{5}{9} \]

Step 6: Solve for x by Taking the Square Root

Take the square root of both sides: \[ x - \frac{1}{3} = \pm \sqrt{\frac{5}{9}} \] Simplifying the right side gives \( x - \frac{1}{3} = \pm \frac{\sqrt{5}}{3} \).

Step 7: Solve for x

Add \(\frac{1}{3}\) to both sides to find \(x\): \[ x = \frac{1}{3} \pm \frac{\sqrt{5}}{3} \] This results in two solutions: \[ x = \frac{1 + \sqrt{5}}{3} \] and \[ x = \frac{1 - \sqrt{5}}{3} \].

Key Concepts

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). These equations are called 'quadratic' because the highest degree term, \(x^2\), is a square. Quadratic equations often model physical phenomena and can be visualized as parabolas when graphed. The solutions to these equations are the values of \(x\) that make the equation true, also known as the roots of the quadratic. Understanding how to manipulate and solve these types of equations is fundamental in algebra.

Quadratic equations can be solved using various methods such as factoring, using the quadratic formula, or, as in this exercise, by completing the square. Each of these methods relies on understanding how quadratic equations are structured and what the solutions represent.

Methods for Solving Equations

There are several methods to solve quadratic equations, and choosing the right one depends on the specific equation you're working with.

• **Factoring**: This method involves rewriting the quadratic as a product of binomials when possible. This approach is straightforward when equations are simple and easily factorable.
• **Quadratic Formula**: For any quadratic equation \(ax^2 + bx + c = 0\), the quadratic formula provides solutions: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It’s a reliable method when factoring is difficult.
• **Completing the Square**: This method involves rearranging the equation into a perfect square trinomial to make solving for \(x\) easier. It's especially useful for equations that are hard to factor.

Completing the square is highlighted in the exercise, where you transform the equation into a perfect square and then use it to find the roots. This method provides insight into the structure of quadratic equations and helps in understanding the derivation of the quadratic formula.

Forming a Perfect Square

A perfect square trinomial is an expression that can be written in the form \((x + p)^2\). The process of completing the square involves modifying a quadratic equation so that the left side becomes a perfect square trinomial. Here's how it works:

Take the coefficient of the \(x\) term, half it, then square it.
For example, in the equation \(x^2 - \frac{2}{3}x = \frac{4}{9}\), taking half of the \(\frac{-2}{3}\) gives \(\frac{-1}{3}\). Squaring this results in \(\frac{1}{9}\). Add \(\frac{1}{9}\) to both sides to form a perfect square trinomial: \(x^2 - \frac{2}{3}x + \frac{1}{9} = \left(x - \frac{1}{3}\right)^2\).

This transformation allows us to express the quadratic equation as a square, simplifying further steps in solving the equation.

Square Root Method for Solving

The square root method involves solving an equation by taking the square root of both sides once you have formed a perfect square. In the exercise above, after forming the perfect square \(\left(x - \frac{1}{3}\right)^2 = \frac{5}{9}\), you apply this method to isolate \(x\).

Here's how it plays out:

• First, take the square root of both sides so that \(x - \frac{1}{3} = \pm \sqrt{\frac{5}{9}}\).
• Simplify the square root to get \(x - \frac{1}{3} = \pm \frac{\sqrt{5}}{3}\).
• Finally, solve for \(x\) by isolating it, which involves adding \(\frac{1}{3}\) to each part: \(x = \frac{1}{3} \pm \frac{\sqrt{5}}{3}\).

This method results in two potential solutions due to the \(\pm\) sign, reflecting the two axes intersections of a parabola—a characteristic of quadratic functions. The square root method, following completion of the square, offers a direct path to finding these solutions.