Problem 59 - Algebra and Trigonometry

Question

Find all solutions of the equation, and express them in the form \(a+b i\) $$ x^{2}-4 x+5=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 2 + i\) and \(x = 2 - i\).

Step 1: 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 \(c = 5\).

Step 2: 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 \(D = 16 - 20 = -4\).

Step 3: Analyze the discriminant

Since the discriminant \(D = -4\) is negative, the quadratic equation has two complex conjugate solutions.

Step 4: 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: \(x = \frac{-(-4) \pm \sqrt{-4}}{2 \times 1} = \frac{4 \pm \sqrt{-4}}{2}\).

Step 5: Simplify the square root of the negative discriminant

The square root of \(-4\) is \(2i\). Thus, \(x = \frac{4 \pm 2i}{2}\).

Step 6: Calculate the solutions

Simplify the expression \(x = \frac{4 \pm 2i}{2}\), which gives the solutions \(x = 2 + i\) and \(x = 2 - i\).

Key Concepts

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:

• \( a = 1 \)
• \( b = -4 \)
• \( c = 5 \)

These coefficients are crucial for solving the quadratic equation, especially when using formulas like the discriminant and the quadratic formula.

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:

• \( D = (-4)^2 - 4 \times 1 \times 5 \)
• \( D = 16 - 20 \)
• \( D = -4 \)

The discriminant tells us that since \( D = -4 \), which is negative, the solutions for this equation will be complex numbers rather than real ones.

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:

• \( a = 1 \)
• \( b = -4 \)
• \( D = -4 \)

This gives us:

• \( x = \frac{-(-4) \pm \sqrt{-4}}{2 \times 1} = \frac{4 \pm \sqrt{-4}}{2} \)

Simplifying further, solve the equations inside the formula, including the square root of \(-4\), which leads us to potential complex solutions.

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:

• \( x = \frac{4 \pm 2i}{2} \)

Breaking this down, we find two complex solutions:

• \( x = 2 + i \)
• \( x = 2 - i \)

These are examples of complex conjugate numbers. Complex solutions are typically expressed in the form \( a + bi \), which are used extensively in various fields of mathematics and engineering.