Question

Solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.\(8 x^2+4 x+5=0\)

Short answer

The solutions to the quadratic equation are given by \(x = -\frac{2}{8} \pm\frac{\sqrt{-14}}{8}\) or \(x = -\frac{1}{4} \pm \frac{\sqrt{19}}{4i}\). Both methods yield these solutions, but the Quadratic Formula method requires fewer steps and computations, so it may be preferred despite producing the same results.

Step-by-step solution

Using the Quadratic Formula

The Quadratic Formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In the given equation, \(a=8\), \(b=4\), and \(c=5\). Substituting these values into the formula, one would compute \(x = \frac{-4 \pm \sqrt{4^2 - 4*8*5}}{2*8}\). Evaluating the expression, and simplifying, one gets \(x = \frac{-2\pm\sqrt{-14}}{8}\).

Solving Using Completing the Square

One could also solve this quadratic equation by completing the square. First, divide the equation by 8 in order to get the coefficient of \(x^2\) as 1. The equation becomes \(x^2 + \frac{1}{2}x + \frac{5}{8} = 0\). Then take half the coefficient of x, square it and subtract it from both sides of the equation to complete the square. The equation becomes \((x + \frac{1/4})^2 = -\frac{19}{32}\). Taking square roots of both sides we also get \(x = -\frac{1}{4} \pm \frac{\sqrt{19}}{4i}\).

Comparing the Methods and Choosing a Preference

Both of the methods bring us to essentially the same solutions, but the Quadratic Formula is generally more straightforward and quicker to use, especially in the case of more difficult problems. Therefore, one might prefer the Quadratic Formula in this case. Completing the square, while effective, may require more steps and thus have a bit more room for error in manual calculations.

Key concepts

Completing the Square

Completing the square is a useful algebraic technique that enables us to solve quadratic equations. It involves rearranging a quadratic equation into a perfect square trinomial, allowing us to find solutions with ease.

To complete the square, it's best to first ensure the coefficient of the quadratic term, usually denoted as \(x^2\), is 1. This sometimes requires dividing the entire equation by this coefficient if it is not already 1. For example, in the equation \(8x^2 + 4x + 5 = 0\), dividing through by 8 was necessary.

Once the equation is in the form of \(x^2 + bx + c = 0\), the next step is to focus on the linear term (the term with \(x\)). Take half of the coefficient of this term, square it, and add it to both the left and right sides of the equation. This creates a perfect square trinomial.

• In the example, \((x + \frac{1}{4})^2\) was formed by taking half of \(\frac{1}{2}\), then squaring it.

This results in the equation being rewritten in the form \( (x + m)^2 = n \), where \(m\) and \(n\) are constants.

Finally, solve for \(x\) by taking the square root of both sides and simplifying, bearing in mind that the solution will include both a positive and negative root.

Quadratic Equation

Quadratic equations are central to algebra and appear in various forms, generally recognizable by the presence of the quadratic term \(x^2\). The standard form of a quadratic equation is given by \(ax^2 + bx + c = 0\). Here, \(a, b,\) and \(c\) are constants with \(a\) not equal to zero.

Solving quadratic equations can be done using several methods, including the Quadratic Formula, factoring, or completing the square, as demonstrated in the example problem. When one encounters such an equation as \(8x^2 + 4x + 5 = 0\), the goal is to find the values of \(x\) that satisfy the equation.

The Quadratic Formula provides a quick and reliable way to find these solutions when factoring is difficult or impossible. The formula is:

• \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

In the provided equation, substituting the values \(a = 8\), \(b = 4\), and \(c = 5\) directly into the formula helps find the solutions efficiently.

Each solution obtained could be real or complex, depending on the value under the square root sign, known as the discriminant. This discriminant (\[-14\]) indicates whether the roots are real (positive) or complex (negative).

Complex Numbers

Complex numbers come into play in quadratic equations when the discriminant is negative, as seen in the example \(\sqrt{-14}\). These numbers consist of a real part and an imaginary part, usually expressed in the form \(a + bi\), where \(i\) is the square root of -1.

In completing the square, or using the Quadratic Formula, when you encounter a situation where the square root of a negative number is needed, it signifies that the solution will involve complex numbers. Imaginary numbers aid in representing these solutions.

• For instance, the equation resulted in roots like \(-\frac{1}{4} \pm \frac{\sqrt{19}}{4i}\).

Complex numbers aren't just a theoretical concept; they have applications in various fields such as engineering and physics. They help in solving problems where real solutions are insufficient.

Remember, when working with complex solutions, always incorporate both the real and imaginary components, as they represent different dimensions of the solution.