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Magazine Chapter 3: Problem 99
Solve the equations over the complex numbers. \(5 x^{2}+6 x+2=0\)
Short Answer
Expert verified
The roots are \(-0.6 + 0.2i\) and \(-0.6 - 0.2i\).
Step by step solution
01
Identify the Quadratic Equation
The equation given is a quadratic equation in the standard form, \(ax^2 + bx + c = 0\), with \(a = 5\), \(b = 6\), and \(c = 2\).
02
Calculate the Discriminant
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute \(a = 5\), \(b = 6\), and \(c = 2\) into the formula: \(D = 6^2 - 4 \times 5 \times 2\).
03
Evaluate the Discriminant
Calculate the value: \(D = 36 - 40 = -4\). Since the discriminant is negative, the roots will be complex numbers.
04
Use the Quadratic Formula
The quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) can be used to find the roots. Substitute \(b = 6\), \(D = -4\), and \(a = 5\) into the formula: \(x = \frac{-6 \pm \sqrt{-4}}{10}\).
05
Simplify Under the Square Root
Simplify \(\sqrt{-4}\) as \(2i\), where \(i\) is the imaginary unit. Substitute this into the formula to get: \(x = \frac{-6 \pm 2i}{10}\).
06
Simplify the Expression
Separate the real and imaginary parts: \(x = \frac{-6 + 2i}{10}\) and \(x = \frac{-6 - 2i}{10}\). Simplify each part to obtain: \(x = -0.6 + 0.2i\) and \(x = -0.6 - 0.2i\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Equation
A quadratic equation is a second degree polynomial in the form \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, and \( x \) represents an unknown variable. In a quadratic equation:
- \( a \) is the coefficient of \( x^2 \), known as the leading coefficient.
- \( b \) is the coefficient of \( x \), which affects the linear component of the equation.
- \( c \) is the constant term.
Discriminant
The discriminant is a part of the quadratic equation used to determine the nature of its roots. It's found using the formula \( D = b^2 - 4ac \). The value of \( D \) tells us:
- If \( D > 0 \), there are two distinct real roots.
- If \( D = 0 \), there is exactly one real root (or a repeated root).
- If \( D < 0 \), there are no real roots, but two complex roots.
Quadratic Formula
The quadratic formula \( x = \frac{-b \pm \sqrt{D}}{2a} \) is a universal method to find the roots of any quadratic equation. It works irrespective of the nature of the discriminant. Here鈥檚 how it's used:
- Substitute \( b \), \( a \), and the calculated \( D \) into the formula.
- Use the plus-minus symbol (\( \pm \)) to account for the fact there are typically two roots.
- The value under the square root, \( \sqrt{D} \), determines the properties of the roots.
Imaginary Unit
The imaginary unit, denoted as \( i \), is a mathematical concept used to extend the real number system to complex numbers. Defined as \( i = \sqrt{-1} \), it suggests that \( i^2 = -1 \). The inclusion of \( i \) means:
- Any real number multiplied by \( i \) gives an imaginary number.
- Complex numbers take the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( bi \) is the imaginary part.
