Problem 6

Question

In $3-14,$ use the quadratic formula to find the imaginary roots of each equation. $$ 2 x^{2}+2 x+1=0 $$

Step-by-Step Solution

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Answer

The imaginary roots of the equation are $-\frac{1}{2} \pm \frac{i}{2}$.

1Step 1: Identify the coefficients

The quadratic equation given is $2x^2 + 2x + 1 = 0$. Identify the coefficients: $a = 2$, $b = 2$, and $c = 1$.

2Step 2: Write the quadratic formula

The quadratic formula is used to solve equations of the form $ax^2 + bx + c = 0$ and is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

3Step 3: Calculate the discriminant

Calculate the discriminant $b^2 - 4ac$ to determine the nature of the roots. Substitute the values $b = 2$, $a = 2$, and $c = 1$: \[b^2 - 4ac = 2^2 - 4(2)(1) = 4 - 8 = -4\]. Since the discriminant is negative, the roots are imaginary.

4Step 4: Substitute into the quadratic formula

Substitute $a = 2$, $b = 2$, and $c = 1$ into the quadratic formula: \[x = \frac{-2 \pm \sqrt{-4}}{4}\].

5Step 5: Simplify the expression

Simplify $\sqrt{-4}$ to $2i$, where $i$ is the imaginary unit. Thus, the values of $x$ are \[x = \frac{-2 \pm 2i}{4}\].

6Step 6: Obtain the simplified roots

Separate the expression into two parts: \[x = \frac{-2}{4} \pm \frac{2i}{4}\], simplifying gives $x = -\frac{1}{2} \pm \frac{i}{2}$.

Key Concepts

Quadratic FormulaDiscriminantImaginary NumbersSimplifying Complex Numbers

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation, which is any equation in the form of $ax^2 + bx + c = 0$. The formula itself is expressed as:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This means you can find the solutions for $x$ by substituting the coefficients $a$, $b$, and $c$ into the formula. The "$\pm$" symbol in the formula indicates there are generally two solutions: one involving addition, the other subtraction. Often, these solutions represent the points where the parabola defined by the quadratic equation crosses the x-axis. However, when dealing with imaginary roots, the curve does not actually intersect the x-axis.

Discriminant

The discriminant is a crucial component in determining the nature of the roots of a quadratic equation. It is found within the quadratic formula, specifically inside the square root, $b^2 - 4ac$. Here’s what it tells us:

If the discriminant is positive, the quadratic equation has two distinct real roots.
If the discriminant is zero, there is one unique real root (also known as a repeated or double root).
If the discriminant is negative, it signals that the equation has two complex or imaginary roots.

In our example equation, $2x^2 + 2x + 1 = 0$, the discriminant calculated as $-4$ indicates imaginary roots, meaning the quadratic does not cross the real x-axis.

Imaginary Numbers

Imaginary numbers are a fascinating part of mathematics, especially when working with quadratic equations. The term "imaginary" refers to numbers that result from the square root of a negative number, which isn’t possible to compute among real numbers. We use the symbol $i$ to denote $\sqrt{-1}$. Thus, any imaginary number can be expressed as a real number multiplied by $i$.
In problems like the one you're working on,

When the discriminant is negative, you’ll end up with a square root of a negative number, which leads to an imaginary number.

In the equation $2x^2 + 2x + 1 = 0$, the square root of $-4$ simplifies to $2i$, resulting in imaginary solutions.

Simplifying Complex Numbers

Once you establish that a quadratic equation has imaginary roots, simplifying involves handling complex numbers effectively. A complex number has the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit.