Problem 61 - A Graphical Approach to College Algebra

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = 1 \pm \sqrt{3}\).

Step 1: Move Constant to the Right Side

We start with the original equation: \(x^2 - 2x = 2\). In order to complete the square, we need to move the constant term to the right side of the equation, so subtract 2 from both sides: \(x^2 - 2x - 2 = 0\). We rewrite it as \(x^2 - 2x = 2\). This means the equation is already ready for completing the square.

Step 2: Complete the Square

To complete the square, take half of the coefficient of \(x\), which is \(-2\), divide it by 2 to get \(-1\), and then square it to get \(1\). Add \(1\) to both sides of the equation: \(x^2 - 2x + 1 = 2 + 1\). This creates a perfect square trinomial on the left side: \((x - 1)^2 = 3\).

Step 3: Solve the Equation

Take the square root of both sides to solve for \(x\): \(x - 1 = \pm \sqrt{3}\). This gives us two possible equations: \(x - 1 = \sqrt{3}\) and \(x - 1 = -\sqrt{3}\).

Step 4: Find Solutions for x

Solve each equation from the previous step: \(x - 1 = \sqrt{3}\) results in \(x = 1 + \sqrt{3}\) and \(x - 1 = -\sqrt{3}\) results in \(x = 1 - \sqrt{3}\). These are the solutions to the original equation.

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions where the highest exponent of the variable is a square. These equations typically look like: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) represent constants, and \( x \) is the variable that you want to solve for.

• The standard form is crucial as it provides a foundation for methods like completing the square.
• Knowing how to manipulate the equation into this form is the first step in solving it.

In the equation from the exercise, \( x^2 - 2x = 2 \), it's already nearly in this form, but the constant needs to be positioned on the other side.
Let's review what this means practically: you rearrange the equation so that it becomes \( x^2 + bx = c \), making it ready for techniques like completing the square.
By grasping this reorganization, you're primed for solving it efficiently.

Perfect Square Trinomial

Completing the square is key in forming a perfect square trinomial. This technique is about reshaping quadratic equations so they're easy to solve by taking advantage of patterns.

• This involves using the middle term's coefficient, dividing it by two, squaring it, and then incorporating it into the equation.

For example, in the equation \( x^2 - 2x = 2 \), we focus on \(-2\), divide it by 2 to get \(-1\), then square it to add \(1\) on both sides.
By performing these steps, we shift the equation to a convenient form: \( (x-1)^2 = 3 \).
What we have now is a perfect square trinomial, \( x^2 - 2x + 1 \). This restructure makes it easier to progress with solving the equation, as it can be expressed as the square of a binomial, like \( (x - 1)^2 \).
Understanding this pattern is foundational in cracking quadratic challenges.

Square Root Property

The square root property is a powerful method for solving equations that have been converted into perfect squares. Once an expression is in a square form, applying this property allows us to find solutions easily.
In our example \((x-1)^2 = 3\), the next step is to rid the square by taking the square root of both sides.

• The square root of a square simply "undoes" the square, therefore \( \sqrt{(x-1)^2} = x-1 \).
• This operation yields two possible values due to the nature of square roots: \( x - 1 = \sqrt{3} \) and \( x - 1 = -\sqrt{3} \).

Solving these separate equations by adding 1 to each side gives the final solutions: \( x = 1 + \sqrt{3} \) and \( x = 1 - \sqrt{3} \).
It’s important to remember both the positive and negative roots since quadratic equations typically have two solutions.
This property simplifies what could otherwise be a complex process, making it straightforward to find your answers.