Problem 28 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

The solution to the quadratic equation \(2x^2-4x-2=0\) by completing the square is \(x=1\)

Step 1: Write the equation in standard form

First, it's best to rewrite the given equation in the standard form of quadratic equation, which is \(ax^2+bx+c=0\). Now divide each term by 2 to simplify, which gives \(x^2-2x-1=0\).

Step 2: Complete the square

To complete the square, the equation is rearranged as \((x^2-2x)=-1\). Next, add the square of half of the coefficient of x, which is \((-2/2)^2=1\), to both sides of the equation. This gives \((x^2-2x+1)=-1+1=0\). The left hand side of the equation forms a perfect square and is written as \((x-1)^2=0\).

Step 3: Solve for the variable x

Finally, solve for x by taking the square root of both sides of the equation. This gives \(x-1=0\) and thus \(x=1\)

Key Concepts

Quadratic Equations

Quadratic equations are simply equations of the second degree, typically in the form \(ax^2 + bx + c = 0\). Understanding the structure is crucial to solving them. Here are the basic terms used in these equations:

• \(a\), \(b\), and \(c\) are constants, with \(a\) never being zero.
• \(x\) represents an unknown variable that we strive to solve.
• The highest power of \(x\) is 2, which is what makes it a quadratic equation.

Quadratic equations can represent various real-life situations, such as projectile motion. Solving them can provide insights into the minimum and maximum values and help estimate roots, intersections, and other critical points. The equation we are examining, \(2x^2 - 4x - 2 = 0\), fits this description and can be systematically solved using different methods.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It provides a simple way to express complex relationships and solve equations, such as quadratic equations. Here are some algebraic principles crucial to solving our example problem:

• Standard Form: The quadratic equation is rearranged to \(ax^2 + bx + c = 0\) to make it easier to handle.
• Simplification: In our problem, the equation \(2x^2 - 4x - 2 = 0\) is simplified by dividing all terms by 2, resulting in \(x^2 - 2x - 1 = 0\). This step makes calculations more manageable.
• Rearrangement: Completing the square requires rearranging terms to facilitate square formation, as in \((x^2 - 2x) = -1\).

By understanding these algebraic basics, we can systematically approach solving quadratic equations using various techniques, aiding in our mastery of algebra.

Solving Equations

Solving equations, especially quadratic ones like \(x^2 - 2x - 1 = 0\), involves specific techniques that transform the equation into a form where the solution can be readily obtained. For example, completing the square is a robust method:

• Identifying Components: Recognize the components required to turn part of the equation into a perfect square, such as taking half of the coefficient of \(x\) (i.e., \(-2/2\)) and squaring it \((-1)^2 = 1\).
• Forming a Perfect Square: Add this squared value to both sides, transforming the equation into a perfect square trinomial, \((x-1)^2 = 0\).
• Finding the Solution: Taking the square root of both sides dissolves the square, leaving us with a linear equation \(x - 1 = 0\). This straightforwardly leads us to the solution \(x = 1\).

This method not only simplifies calculations but also provides a clear pathway from a complex equation to its solution. Mastery of these steps is essential for efficiently solving quadratic equations.