Problem 34 - Introductory Algebra for College Students

Question

Solve each quadratic equation using the quadratic formula. $$6 x^{2}=-2 x-1$$

Step-by-Step Solution

Answer

The solutions to the given quadratic equation are \(-\frac{1}{6} ± \frac{\sqrt{5}}{6}i\).

Step 1: Re-arrange the Equation

The first step is to rearrange the given equation to be in standard form. This is done by adding \(2x + 1\) to both sides to get \(6x^{2} + 2x + 1 = 0\).

Step 2: Identify Coefficients

In the equation \(6x^{2} + 2x + 1 = 0\), identify the coefficients as \(a=6\), \(b=2\), and \(c=1\).

Step 3: Apply Quadratic Formula

Use these coefficients in the quadratic formula \(\frac{-b ± \sqrt{b^2 - 4ac}}{2a}\). Substitute the identified values \(a=6\), \(b=2\), and \(c=1\) into the formula to get the solutions \(\frac{-2 ± \sqrt{(2)^2 - 4*6*1}}{2*6}\).

Step 4: Calculate the Discriminant

The value inside the square root, called the discriminant, is \(b^{2} - 4ac\). Compute this as \((2)^2 - 4*6*1 = 4 - 24 = -20 \). In this case, the discriminant is negative, which means that solutions will be a pair of complex numbers.

Step 5: Calculate the Solutions

Now, calculate the solutions by using the value of the discriminant in the quadratic formula. The two solutions are \(\frac{-2 ± \sqrt{-20}}{12}\), simplifying to \(-\frac{1}{6} ± \frac{\sqrt{20}}{12}i\). In simplified form, solutions are \(-\frac{1}{6} ± \frac{\sqrt{5}}{6}i\).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful mathematical tool used to find the solutions of quadratic equations. A quadratic equation typically takes the form of \[ax^2 + bx + c = 0\], where \(a\), \(b\), and \(c\) are the equation's coefficients. The solutions or roots of this equation can be found using the quadratic formula:

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

This equation provides the values of \(x\) that satisfy the quadratic equation. The "\(\pm\)" symbol in the formula indicates that there may be two possible solutions: one for the plus and one for the minus.
The quadratic formula is applicable regardless of the nature of the coefficients or the discriminant, making it a versatile method for solving quadratic equations. However, the type of the solutions (real or complex) is determined by the value of the discriminant, which we will discuss next.

Complex Numbers

Complex numbers come into play when the solutions to a quadratic equation are not real numbers. This situation occurs when the discriminant (the expression under the square root in the quadratic formula) is negative. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
The imaginary unit \(i\) has the property that \(i^2 = -1\).
When dealing with a negative discriminant, the solutions to the quadratic equation will involve \(i\). For example, in our specific case, the expression \(\sqrt{-20}\) simplifies to \(2\sqrt{5}i\), giving us solutions involving complex numbers.

• Complex solutions usually appear in conjugate pairs: \(a + bi\) and \(a - bi\).

Understanding complex numbers is crucial, as they are key to solving quadratic equations with negative discriminants.

Discriminant

The discriminant is a critical component of the quadratic formula and is represented by the expression \(b^2 - 4ac\). It determines the nature and number of the solutions to a quadratic equation.
Here's how it works:

• If \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is exactly one real solution, often called a repeated or double root.
• If \(b^2 - 4ac < 0\), the equation has two complex solutions, which are conjugates of each other.

In our example, the discriminant was calculated as \(-20\). Since it is negative, this confirms the presence of complex number solutions. The nature of the discriminant helps us predict whether the quadratic equation can have real solutions or if we should expect complex numbers instead.

Algebraic Coefficients

The algebraic coefficients \(a\), \(b\), and \(c\) in a quadratic equation \(ax^2 + bx + c = 0\) play a vital role in solving the equation using the quadratic formula. These coefficients are used directly in the formula and affect the calculation of the discriminant as well.
Determining these coefficients correctly is the first essential step in the solution process.

• \(a\) is the coefficient of \(x^2\),
• \(b\) is the coefficient of \(x\),
• \(c\) is the constant term with no \(x\) attached.

For our problem, the coefficients were identified as \(a = 6\), \(b = 2\), and \(c = 1\). Using these values accurately in the quadratic formula is crucial for arriving at the correct solutions. Knowing how to extract and utilize these coefficients enhances your ability to solve quadratic equations effectively.