Problem 59 - Precalculus Mathematics for Calculus

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 Equation

The equation given is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -4 \), and \( c = 5 \).

Step 2: Calculate the Discriminant

The discriminant \( \Delta \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). Plugging in the values, we get \( \Delta = (-4)^2 - 4 \times 1 \times 5 = 16 - 20 = -4 \).

Step 3: Determine the Nature of the Roots

Since the discriminant \( \Delta = -4 \) is negative, the equation has complex roots. We will use the quadratic formula to find these roots.

Step 4: Apply the Quadratic Formula

The roots of the quadratic equation can be found using the formula \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \). Substituting \( a = 1 \), \( b = -4 \), and \( \Delta = -4 \) into the formula gives \( x = \frac{-(-4) \pm \sqrt{-4}}{2 \times 1} \).

Step 5: Simplify the Expression

Calculate \( \sqrt{-4} \) as \( 2i \). Thus, the roots are \( \frac{4 \pm 2i}{2} \). Simplify this to find the solutions: \( x = 2 \pm i \).

Step 6: Express in Form \( a+bi \)

The two solutions of the equation are \( x = 2 + i \) and \( x = 2 - i \), both of which are in the required form \( a + bi \).

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental mathematical expression, taking the general form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Such equations are vital in both pure and applied mathematics because they can model a wide variety of phenomena. The value \( a \) determines the width and direction of the parabola represented by the equation graphically, while \( b \) controls the symmetry axis of the parabola, and \( c \) signifies the y-intercept.

• The parabola opens upwards if \( a > 0 \) and downwards if \( a < 0 \).
• The solutions of the quadratic equation are where this parabola intersects the x-axis, if they exist as real solutions.
• A quadratic equation may have two real solutions, one real solution, or two complex solutions, depending on the value of the discriminant.

Understanding the structure and components of the quadratic equation helps solve it effectively, revealing important implications in various scientific or engineering contexts.

Discriminant

The discriminant is a determining factor for the nature of the roots of a quadratic equation. It is symbolized by \( \Delta \) and is calculated using the formula \( \Delta = b^2 - 4ac \). The value of the discriminant gives us crucial insight into the type of roots we can expect from the equation.

• If \( \Delta > 0 \), the equation has two distinct real roots. This indicates two points where the parabola crosses the x-axis.
• If \( \Delta = 0 \), there is exactly one solution, known as a repeated or double root, where the parabola touches the x-axis without crossing it.
• If \( \Delta < 0 \), as in the case of the equation \( x^2 - 4x + 5 = 0 \), the roots are complex, and the parabola does not intersect the x-axis at any real point.

By analyzing the discriminant, one can predict the nature of the roots even before solving the equation, which aids in selecting the most suitable method for finding these roots.

Quadratic Formula

The quadratic formula is a universal method to find solutions for any quadratic equation. It is given as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), and is particularly useful when the equation does not readily factor or when the discriminant is not a perfect square.

• The term under the square root, \( b^2 - 4ac \), is the discriminant and it dictates whether the roots are real or complex.
• The formula provides two solutions: \( x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a} \) and \( x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a} \).
• For the equation \( x^2 - 4x + 5 = 0 \), substituting \( a = 1 \), \( b = -4 \), and \( c = 5 \), we found that the solutions are complex: \( x = 2 \pm i \).

Employing the quadratic formula, especially with a negative discriminant as shown, is vital for solving equations efficiently and provides an elegant way to express complex solutions.