Problem 31 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(2 x^{2}+10 x+11=0\)

Step-by-Step Solution

Answer

The solution to the quadratic equation is \(x = -\frac{5}{2} \pm \frac{1}{2}i\)

Step 1: Make the Coefficient of \(x^2\) One

To start, the equation is divided by 2 to make the coefficient of \(x^2\) equal to one. The equation becomes: \(x^{2} + 5x + \frac{11}{2}=0\)

Step 2: Complete the Square

Next, to complete the square, take half of the coefficient of \(x\), square it, and add it to both sides of the equation. We have to subtract the same value from the right side of the equation to balance it out. So, \(x^{2} + 5x + \left(\frac{5}{2}\right)^2 = -\frac{11}{2} + \left(\frac{5}{2}\right)^2\). This simplifies to \((x + \frac{5}{2})^2 = \frac{-1}{4}\)

Step 3: Square Root Property

Lastly apply the square root property. Take the square root of both sides and solve for \(x\). This leads to \(x + \frac{5}{2} = \pm \sqrt{\frac{-1}{4}}\). By simplifying, we get \(x = -\frac{5}{2} \pm \frac{1}{2}i\) where \(i=\sqrt{-1}\).

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). Solving these equations involves finding the values of \(x\) that satisfy the equation.
A common method used to solve quadratic equations is the "completing the square" technique. This involves manipulating the equation to form a perfect square trinomial, which can then be solved using the square root property.

• Start by ensuring the coefficient of \(x^2\) is one. If not, divide the entire equation by \(a\).
• Rearrange the equation if necessary so that terms involving \(x\) are on one side of the equation.
• Add and subtract the square of half the \(x\)-coefficient to complete the square, thereby transforming the equation into \((x + p)^2 = q\).

Completing the square is particularly useful when the quadratic equation does not factor easily. It provides a direct way to transform the equation into a recognizable perfect square form that can be easily solved.

Complex Numbers

Complex numbers introduce an elegant solution to quadratic equations that do not have real roots. These numbers are of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which is defined as \(i = \sqrt{-1}\).
Imaginary numbers come into play when you take the square root of a negative number, which is exactly what happens in the given problem.

• When we solve the quadratic equation by completing the square, we end up with \((x + \frac{5}{2})^2 = \frac{-1}{4}\).
• The negative under the square root sign signifies that we are dealing with complex numbers.
• This results in solutions \(x = -\frac{5}{2} \pm \frac{1}{2}i\).

Complex numbers can be visualized as a point in the complex plane, where the x-axis represents the real part (\(a\)) and the y-axis represents the imaginary part (\(bi\)). They expand our number system beyond real numbers and are crucial in fields like engineering and physics.

Square Root Property

The square root property is a key technique used in solving quadratic equations that have been simplified to the form \((x + p)^2 = q\). Once the equation is in this form, you can proceed by taking the square root of both sides.
Here’s how you use this property effectively:

• Isolate the squared term: Ensure the equation is structured as \((x + p)^2 = q\).
• Take the square root of both sides: Apply the square root property to both sides, keeping in mind that each side will yield two possible values, plus and minus: \(x + p = \pm \sqrt{q}\).
• Solve for \(x\): Take into account both the positive and negative roots to find potential solutions for \(x\).

It's important to remember that the square root of a negative number will involve an imaginary number, as seen in the example problem, where \(\sqrt{-1} = i\). This application of the square root property allows us to handle both real and complex solutions seamlessly.