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Magazine Chapter 5: Problem 6
In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ 2 x^{2}+2 x+1=0 $$
Short Answer
Expert verified
The imaginary roots of the equation are \(-\frac{1}{2} \pm \frac{i}{2}\).
Step by step solution
01
Identify the coefficients
The quadratic equation given is \(2x^2 + 2x + 1 = 0\). Identify the coefficients: \(a = 2\), \(b = 2\), and \(c = 1\).
02
Write the quadratic formula
The quadratic formula is used to solve equations of the form \(ax^2 + bx + c = 0\) and is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
03
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.
04
Substitute into the quadratic formula
Substitute \(a = 2\), \(b = 2\), and \(c = 1\) into the quadratic formula: \[x = \frac{-2 \pm \sqrt{-4}}{4}\].
05
Simplify the expression
Simplify \(\sqrt{-4}\) to \(2i\), where \(i\) is the imaginary unit. Thus, the values of \(x\) are \[x = \frac{-2 \pm 2i}{4}\].
06
Obtain the simplified roots
Separate the expression into two parts: \[x = \frac{-2}{4} \pm \frac{2i}{4}\], simplifying gives \(x = -\frac{1}{2} \pm \frac{i}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
- \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
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鈥檚 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.
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鈥檛 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,
In problems like the one you're working on,
- When the discriminant is negative, you鈥檒l end up with a square root of a negative number, which leads to an imaginary number.
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.
- For example, let鈥檚 consider \(x = \frac{-2 \pm 2i}{4}\).
- Start by separating the real and the imaginary components: \(\frac{-2}{4} \pm \frac{2i}{4}\).
- Simplify each part: \(x = -\frac{1}{2} \pm \frac{i}{2}\).
