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Magazine Chapter 1: Problem 59
Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}-4 x+5=0 $$
Short Answer
Expert verified
The solutions are \(x = 2 + i\) and \(x = 2 - i\).
Step by step solution
01
Identify the quadratic equation format
We have the quadratic equation \(x^2 - 4x + 5 = 0\). The standard form is \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = -4\), and \(c = 5\).
02
Calculate the discriminant
The discriminant \(D\) is given by \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) to get \(D = (-4)^2 - 4 \times 1 \times 5\). This simplifies to \(D = 16 - 20 = -4\).
03
Analyze the discriminant
Since the discriminant \(D = -4\) is negative, the quadratic equation has two complex conjugate solutions.
04
Use the quadratic formula
The quadratic formula is \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute \(a = 1\), \(b = -4\), and \(D = -4\) into the formula: \(x = \frac{-(-4) \pm \sqrt{-4}}{2 \times 1} = \frac{4 \pm \sqrt{-4}}{2}\).
05
Simplify the square root of the negative discriminant
The square root of \(-4\) is \(2i\). Thus, \(x = \frac{4 \pm 2i}{2}\).
06
Calculate the solutions
Simplify the expression \(x = \frac{4 \pm 2i}{2}\), which gives the solutions \(x = 2 + i\) and \(x = 2 - i\).
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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 polynomial equation of degree 2. Its standard form is represented as \( ax^2 + bx + c = 0 \). Here, "a," "b," and "c" are constants, with "a" not being equal to zero, since that would make it a linear equation instead.
For our specific problem, the quadratic equation given is \( x^2 - 4x + 5 = 0 \). By comparing it with the standard form, we identify:
For our specific problem, the quadratic equation given is \( x^2 - 4x + 5 = 0 \). By comparing it with the standard form, we identify:
- \( a = 1 \)
- \( b = -4 \)
- \( c = 5 \)
Discriminant
The discriminant of a quadratic equation is a value that can reveal important information about the nature of the roots of the equation. The formula for the discriminant \( D \) is given by \( D = b^2 - 4ac \). This is derived from the standard form of the quadratic equation.
For the equation \( x^2 - 4x + 5 = 0 \), let's substitute \( a = 1 \), \( b = -4 \), and \( c = 5 \) into the discriminant formula:
For the equation \( x^2 - 4x + 5 = 0 \), let's substitute \( a = 1 \), \( b = -4 \), and \( c = 5 \) into the discriminant formula:
- \( D = (-4)^2 - 4 \times 1 \times 5 \)
- \( D = 16 - 20 \)
- \( D = -4 \)
Quadratic Formula
The quadratic formula is a universal method for finding the solutions of any quadratic equation. It is stated as \( x = \frac{-b \pm \sqrt{D}}{2a} \), where \( D \) represents the discriminant. The "+" and "-" in the formula stands for the two possible solutions, arising from the nature of quadratic functions.
Using the quadratic formula for our equation, we substitute:
Using the quadratic formula for our equation, we substitute:
- \( a = 1 \)
- \( b = -4 \)
- \( D = -4 \)
- \( x = \frac{-(-4) \pm \sqrt{-4}}{2 \times 1} = \frac{4 \pm \sqrt{-4}}{2} \)
Complex Solutions
Complex solutions arise from quadratic equations when the discriminant is negative. A negative discriminant implies that the roots are not real numbers. Instead, they involve the imaginary unit \( i \), where \( i = \sqrt{-1} \).
For our quadratic equation, we found the discriminant to be \( D = -4 \). The square root of \(-4\) is \( 2i \), and substituting this into our quadratic formula, we get:
For our quadratic equation, we found the discriminant to be \( D = -4 \). The square root of \(-4\) is \( 2i \), and substituting this into our quadratic formula, we get:
- \( x = \frac{4 \pm 2i}{2} \)
- \( x = 2 + i \)
- \( x = 2 - i \)
