Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Find the solutions of the equation $$ x^{2}-5 x+20=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{5}{2} + \frac{i\sqrt{55}}{2} \) and \( x = \frac{5}{2} - \frac{i\sqrt{55}}{2} \).

Step 1: Identify the form of the equation

The equation given is a quadratic equation in the form of \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -5 \), and \( c = 20 \). We will use the quadratic formula to find the solutions.

Step 2: Write the quadratic formula

The quadratic formula used to solve equations of the form \( ax^2 + bx + c = 0 \) is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the values of \( a \), \( b \), and \( c \) into this formula.

Step 3: Calculate the discriminant

First, calculate the discriminant \( \Delta = b^2 - 4ac \). In our equation: \( b = -5 \), \( a = 1 \), and \( c = 20 \). Thus, \( \Delta = (-5)^2 - 4 \times 1 \times 20 = 25 - 80 = -55 \).

Step 4: Determine the nature of the roots

Since the discriminant \( \Delta = -55 \) is negative, the equation has two complex roots. Real roots occur when the discriminant is zero or positive.

Step 5: Calculate the real and imaginary parts of the complex roots

For a negative discriminant, use the quadratic formula, incorporating the imaginary unit \( i \): \[ x = \frac{-b \pm i\sqrt{-\Delta}}{2a} \] Substitute \( b = -5 \), \( \Delta = -55 \), and \( a = 1 \) into the formula: \[ x = \frac{5 \pm i\sqrt{55}}{2} \].

Step 6: Express the final solution in standard form

The solutions to the equation are complex numbers: \( x = \frac{5}{2} + \frac{i\sqrt{55}}{2} \) and \( x = \frac{5}{2} - \frac{i\sqrt{55}}{2} \). Which are in the form \( x = a + bi \), where \( a \) and \( b \) are real numbers.

Key Concepts

Quadratic Formula

The quadratic formula is a mathematical tool used to find solutions for quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). This formula is an essential part of algebra, and it provides an effective way to find the roots, or solutions, of any quadratic equation, regardless of the discriminant. The formula itself is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here:

• \(a\), \(b\), and \(c\) are coefficients of the quadratic equation.
• \(-b\) means you take the opposite sign of \(b\).
• The symbol \(\pm\) indicates two possible values due to the presence of both a positive and a negative root.
• The term under the square root, \(b^2 - 4ac\), is called the discriminant.

The quadratic formula automatically takes into account the character of the roots, whether they are real or complex, by incorporating the discriminant in its calculation.Knowing how to use this formula helps in solving any quadratic equation systematically.
The quadratic formula is particularly convenient when the equation is not easily factorable or when working with complex coefficients.

Complex Roots

When working with quadratic equations, understanding complex roots becomes crucial, especially when the discriminant is negative. Complex numbers consist of a real part and an imaginary part and are expressed in the form \(a + bi\). Here, \(i\) is the imaginary unit, defined by \(i^2 = -1\). Quadratic equations with negative discriminants have complex roots. This is because the square root of a negative number does not exist among real numbers.
To solve such equations using the quadratic formula, we rewrite the square root term as \(i\sqrt{-\Delta}\) thus incorporating the imaginary unit:\[x = \frac{-b \pm i\sqrt{-\Delta}}{2a}\]Let's break this down:

• The part \(-b\) still represents the real shift.
• \(i\sqrt{-\Delta}\) represents the imaginary component, reflecting the non-real nature of the roots.
• Complex roots occur in conjugate pairs. If one root is \(a + bi\), then the other is \(a - bi\).

Understanding how to recognize and compute complex roots helps demystify the process and settles the debate on whether a quadratic equation always has solutions (it does, thanks to complex numbers!).

Discriminant Calculation

The discriminant is a vital component in determining the nature of the roots of a quadratic equation. It is represented by the expression \(b^2 - 4ac\) in the quadratic formula. The discriminant helps predict whether the roots are real and distinct, real and identical, or complex.Here's how the discriminant works:

• If \(\Delta > 0\): The equation has two distinct real roots.
• If \(\Delta = 0\): The equation has exactly one real root, also called a repeated or double root.
• If \(\Delta < 0\): The equation has two complex roots, which are non-real.

Calculating the discriminant before applying the quadratic formula can provide valuable insight into what kind of solutions to expect.For the equation \(x^2 - 5x + 20 = 0\), calculating the discriminant \((-5)^2 - 4 \times 1 \times 20 = -55\) indicates complex roots. This insight guides us towards using complex numbers for a complete solution. Simply put, the discriminant is your first step in solving any quadratic equation to understand what type of numbers the solutions will be.