Problem 55 - Intermediate Algebra

Question

For each quadratic equation, first use the discriminant to determine whether the equation has two nonreal complex solutions, one real solution with a multiplicity of two, or two real solutions. Then solve the equation. $$ x^{2}-7 x+13=0 $$

Step-by-Step Solution

Answer

The quadratic equation has two nonreal complex solutions: \(x = \frac{7 + i\sqrt{3}}{2}\) and \(x = \frac{7 - i\sqrt{3}}{2}\).

Step 1: Identify the coefficients

For the given quadratic equation \(x^2 - 7x + 13 = 0\), identify the coefficients \(a\), \(b\), and \(c\). Here, \(a = 1\), \(b = -7\), and \(c = 13\).

Step 2: Calculate the discriminant

The discriminant \(D\) for a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(D = b^2 - 4ac\). Substitute the values into the formula: \(D = (-7)^2 - 4 \times 1 \times 13 = 49 - 52 = -3\).

Step 3: Interpret the discriminant

Since the discriminant \(D = -3\) is less than zero, the quadratic equation has two nonreal complex solutions.

Step 4: Use the quadratic formula

To find the solutions of the quadratic equation \(x^2 - 7x + 13 = 0\), use the quadratic formula: \(x = \frac{-b \pm \sqrt{D}}{2a}\). Substitute the values of \(b = -7\), \(D = -3\), and \(a = 1\) into the formula: \(x = \frac{7 \pm \sqrt{-3}}{2}\).

Step 5: Express the complex solutions

Since \(\sqrt{-3} = i\sqrt{3}\), the solutions can be written as: \(x = \frac{7 \pm i\sqrt{3}}{2}\). Thus, the solutions are \(x = \frac{7 + i\sqrt{3}}{2}\) and \(x = \frac{7 - i\sqrt{3}}{2}\).

Key Concepts

Discriminant

The discriminant is a crucial part of the quadratic equation, especially when determining the nature of its roots. The discriminant, represented by the letter \( D \), is calculated using the formula:

• \( D = b^2 - 4ac \)

Here, \( a \), \( b \), and \( c \) are coefficients from the quadratic equation \( ax^2 + bx + c = 0 \).The value of the discriminant tells us about the roots of the equation:

• If \( D > 0 \), the equation has two distinct real solutions.
• If \( D = 0 \), there is one real solution, sometimes referred to as a double root.
• If \( D < 0 \), the equation has two non-real, complex solutions.

In our example, the discriminant is \( -3 \), indicating that we have complex solutions.This means that the solutions cannot be expressed as real numbers but involve imaginary numbers.

Complex Solutions

When the discriminant of a quadratic equation is negative, the solutions to the equation are not real numbers. Instead, they are complex numbers. Complex numbers are numbers that have a real part and an imaginary part.The imaginary unit is represented by \( i \), which is equivalent to \( \sqrt{-1} \).For a quadratic with a negative discriminant \( D \), the solutions appear in the form \( x = \frac{-b \pm \, i\sqrt{|D|}}{2a} \).In our equation, the discriminant is \( -3 \):

• \( \sqrt{-3} = i\sqrt{3} \) since \( \sqrt{-1} = i \).
• Our solutions are \( x = \frac{7 \pm i\sqrt{3}}{2} \).

These complex solutions imply that we do not cross the x-axis on a graph but instead form a parabola opening upwards without real roots.

Quadratic Formula

The quadratic formula is a reliable tool to solve quadratic equations and find their roots. No matter the values of \( a \), \( b \), and \( c \), the quadratic formula provides a path to the solutions.The formula is given by:

• \( x = \frac{-b \pm \sqrt{D}}{2a} \)

This formula helps to find both real and complex roots of a quadratic equation.To apply it, follow these steps:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \).
• Calculate the discriminant \( D = b^2 - 4ac \).
• Substitute \( b \), \( D \), and \( a \) into the formula.

For the example \( x^2 - 7x + 13 = 0 \):

• The coefficients \( a = 1 \), \( b = -7 \), and \( c = 13 \).
• The discriminant \( D = -3 \).
• Solutions: \( x = \frac{7 \pm i\sqrt{3}}{2} \).

Using the quadratic formula gives a complete solution, important for cases with imaginary numbers.