Problem 32 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(2 x^{2}+8 x+5=0\)

Step-by-Step Solution

Answer

The solutions of the quadratic equation are \(x=-2+\sqrt{1.5} \) and \(x=-2-\sqrt{1.5} \)

Step 1: Normalize the equation

First, since the coefficient of \(x^{2}\) is not 1, it is necessary to divide the entire equation by 2 to normalize it: \( x^{2}+4x+2.5=0 \)

Step 2: Rearrange the equation

Rearrange the equation in the form \((x-a)^{2}=b)\ by completing the square. The process involves halving the coefficient of \(x\), squaring this value and adding to both sides of the equation. Here, halving the coefficient of \(x\) gives 2 and squaring this value results in 4. So, the equation becomes: \(x^{2}+4x+4=4-2.5,\ \(x+2)^{2}=1.5 \)

Step 3: Solve for x

The last step is to solve for \(x\). Do this by taking square roots of both sides and considering both positive and negative values: \(\sqrt{(x+2)^{2}}=\sqrt{1.5},\ x+2=\pm\sqrt{1.5}. \) This results in two possible values for \(x: x=-2+\sqrt{1.5} \ and \ x=-2-\sqrt{1.5} \)

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants and \( a \) is not zero. This equation is called 'quadratic' because of the highest degree, which is 2. The quadratic equation is a fundamental concept in algebra that often models real-world scenarios.

The solutions to a quadratic equation are the values of \( x \) that make the equation true. These solutions are sometimes called the "roots" of the equation. Unlike linear equations, quadratic equations can have zero, one, or two real solutions. To find these solutions, different methods can be used, including factoring, using the quadratic formula, and completing the square, as shown in the exercised provided. Understanding the form and characteristics of quadratic equations is crucial to solving them effectively.

Solve for x

Solving for \( x \) in the context of quadratic equations generally means finding the values of \( x \) that satisfy the equation. In the original exercise, the equation was solved by completing the square. Let's walk through this powerful technique:

• First, ensure the quadratic term's coefficient, \( a \), is 1 or normalize the equation.
• Next, rearrange the equation to begin completing the square, aiming to express it in the form \((x-a)^2 = b\).
• Here, you adjust the equation by halving the linear term coefficient, squaring it, and then adding this square inside the equation.
• Finally, take the square root on both sides and solve for \( x \), remembering to consider both positive and negative solutions.

This clear set of steps highlights the elegant nature of solving quadratic equations by completing the square, providing two potential solutions given by the equation \( x+2 = \pm\sqrt{1.5} \). This shows the flexibility and usefulness of the technique across diverse equations.

Normalizing Equations

Normalizing equations is a crucial first step when solving quadratic equations through specific methods like completing the square. Normalization refers to adjusting the equation so that the coefficient of the squared term \( x^2 \) is 1. This simplifies the process of completing the square and solving the equation.

In the exercise provided, the equation was \( 2x^2 + 8x + 5 = 0 \). The initial step involved dividing the entire equation by 2 to normalize it. This step adjusted all terms, resulting in \( x^2 + 4x + 2.5 = 0 \). By ensuring the equation is in this form, we simplify subsequent calculations, making it easier to complete the square and ultimately solve for \( x \). This method highlights the importance of normalizing equations as a foundation for effective problem-solving in quadratic equations.