Problem 28 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by completing the square. Show each step. $$ x^{2}+\frac{2}{3} x-\frac{1}{3}=0 $$

Step-by-Step Solution

Answer

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

Step 1: Move Constant Term

To complete the square, start by ensuring the quadratic is in the form \( ax^2 + bx = c \). Move the constant to the right side of the equation: \( x^2 + \frac{2}{3}x = \frac{1}{3} \).

Step 2: Divide Coefficient of x

Take half of the coefficient of \(x\), which is \(\frac{2}{3}\), divide it by 2 to get \(\frac{1}{3}\).

Step 3: Square Result

Square the result from step 2, \((\frac{1}{3})^2 = \frac{1}{9}\). Add \(\frac{1}{9}\) to both sides of the equation to maintain equality: \(x^2 + \frac{2}{3}x + \frac{1}{9} = \frac{1}{3} + \frac{1}{9}\).

Step 4: Simplify Right Side

Combine the fractions on the right side of the equation: \(\frac{1}{3} + \frac{1}{9} = \frac{3}{9} + \frac{1}{9} = \frac{4}{9}\). So now the equation is \(x^2 + \frac{2}{3}x + \frac{1}{9} = \frac{4}{9}\).

Step 5: Write as Perfect Square

The left side of the equation \(x^2 + \frac{2}{3}x + \frac{1}{9}\) can be written as \((x + \frac{1}{3})^2\). So, \((x + \frac{1}{3})^2 = \frac{4}{9}\).

Step 6: Solve for x

Take the square root of both sides to find \(x\). \(x + \frac{1}{3} = \pm \frac{2}{3}\).

Step 7: Isolate x

Subtract \(\frac{1}{3}\) from both potential solutions: 1. \(x = \frac{2}{3} - \frac{1}{3} = \frac{1}{3}\). 2. \(x = -\frac{2}{3} - \frac{1}{3} = -1\).

Key Concepts

Completing the Square

Completing the square is a method used to solve quadratic equations. It involves making a trinomial into a perfect square trinomial, which is a polynomial that can be written as the square of a binomial. This method makes it easier to solve the equation by converting it into a simpler form. To start completing the square, ensure the quadratic equation is in the form of \[ax^2 + bx + c = 0\]. Then, move the constant term to the other side of the equation, focusing on the terms involving \(x\).

• Separate the quadratic and linear terms: Aim for something like \(ax^2 + bx = c\). This sets the stage for manipulating the equation into a perfect square.
• Identify the coefficient of \(x\) and make a note of it for the next steps in the process.

Practice makes perfect, so reviewing multiple examples will help solidify understanding.

Solving Quadratics

Solving quadratic equations involves finding the values of \(x\) that make the equation true. Quadratics take the form \[ax^2 + bx + c = 0\]. In completing the square, the process involves transforming the quadratic equation into a format that allows for easier solving.

• After transforming the quadratic into a perfect square, solving it involves initially taking the square root of both sides.
• This leaves us with two potential solutions, as square roots return positive and negative results: If the equation is \(( x + h )^2 = k\), the solutions are \(x + h = \pm\sqrt{k}\).
• Bears in mind, all manipulations should be evenly balanced; whatever is done to one side, do to the other to maintain equality.

Mastering these concepts allows tackling real-life problems mathematically.

Square Roots

Square roots are integral to solving quadratic equations that have been completed to a square. After rewriting the quadratic in the form of a perfect square, the next step often involves taking square roots.

• The square root operation is used to "cancel" the square of the binomial, simplifying the equation substantially.
• Remember, the square root function returns both a positive and a negative value. For instance, \( \sqrt{9} \) is \(3\) and \(-3\).
• Always apply the square root to both sides of the equation to ensure the integrity of the equality is maintained.

Be cautious: while square roots can simplify equations, they must be applied carefully to avoid errors.

Algebraic Manipulation

Algebraic manipulation is the backbone of solving equations, especially quadratics. It refers to using algebraic techniques to rearrange and simplify expressions and equations. When completing the square, these manipulations ensure that the steps lead to a usable equation.

• Keep each step clear: Always ensure the same operations are done to both sides of the equation.
• Pay attention to fractions and coefficients: Accurate manipulation of fractions and coefficients is crucial for a correct solution.
• Simplify at every step: Whenever possible, reduce expressions to their simplest forms to make calculations easier.

Practicing these techniques can lead to more efficient and confident problem-solving.