Problem 5 - Algebra for College Students

Question

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 equation has two complex solutions: \(x = \frac{7 \pm i\sqrt{3}}{2}\).

Step 1: Identify the components of the quadratic equation

The equation is given as \(x^2 - 7x + 13 = 0\). Here, the components are: \(a = 1\), \(b = -7\), and \(c = 13\).

Step 2: Calculate the discriminant

The formula for the discriminant \(\Delta\) is given by \(\Delta = b^2 - 4ac\). Substituting the values from the equation: \[\Delta = (-7)^2 - 4 \times 1 \times 13 = 49 - 52 = -3\].

Step 3: Determine the nature of the roots

Since the discriminant \(\Delta = -3\) is less than zero, this indicates that the equation has two nonreal complex solutions.

Step 4: Solve the equation using the quadratic formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substituting the values: \[x = \frac{-(-7) \pm \sqrt{-3}}{2 \times 1} = \frac{7 \pm \sqrt{-3}}{2}\].

Step 5: Simplify the expression

Since \(\sqrt{-3}\) is an imaginary number, we can express it as \(\sqrt{3}i\). Therefore, the solutions can be simplified to: \[x = \frac{7 \pm i\sqrt{3}}{2}\].

Key Concepts

Discriminant

The discriminant is a vital part of understanding quadratic equations. It is represented by the Greek letter delta, \( \Delta \). The discriminant helps us determine the nature of the solutions to a quadratic equation before solving it fully. For any quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is calculated using the formula: \[ \Delta = b^2 - 4ac \] The result of this calculation can tell us one of three scenarios about the roots of the quadratic equation:

• If \( \Delta > 0 \), there are two distinct real solutions.
• If \( \Delta = 0 \), there is exactly one real solution, but it's a double root, meaning it appears twice.
• If \( \Delta < 0 \), the solutions are two nonreal complex numbers, meaning they have imaginary components.

In our case, for the equation \( x^2 - 7x + 13 = 0 \), the discriminant is \( -3 \). This outcome, being negative (\( \Delta < 0 \)), indicates that our equation has two nonreal complex solutions.

Complex Solutions

When solving quadratic equations, encountering complex solutions means dealing with numbers that include a part that involves the imaginary unit \( i \), where \( i = \sqrt{-1} \). In mathematical terms, complex numbers have two components:

• The real part, which is a regular number as we know it.
• The imaginary part, which is a multiple of \( i \).

A complex solution to a quadratic equation takes the form of \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, that is multiplied by \( i \).For our quadratic equation \( x^2 - 7x + 13 = 0 \), the discriminant was negative, suggesting complex roots. As we calculated, the solutions were given as: \[ x = \frac{7 \pm i\sqrt{3}}{2} \]This simplified expression gives us two solutions:

• \( \frac{7 + i\sqrt{3}}{2} \), which is one complex solution.
• \( \frac{7 - i\sqrt{3}}{2} \), which is the conjugate of the first.

These solutions show how a negative discriminant leads to imaginary components in the solutions.

Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of any quadratic equation, no matter what kind of solutions it has. The formula comes in handy particularly when factoring is difficult or impossible.The quadratic formula is:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here are the steps to using the quadratic formula effectively:

• Identify the coefficients \( a \), \( b \), and \( c \) from the quadratic equation \( ax^2 + bx + c = 0 \).
• Calculate the discriminant \( \Delta = b^2 - 4ac \) to understand the type of solutions you will find.
• Substitute these values into the quadratic formula.
• Simplify the expression to find the roots.

In our example, \( a = 1 \), \( b = -7 \), and \( c = 13 \). Plugging these into the quadratic formula, we find:\[ x = \frac{7 \pm \sqrt{-3}}{2} \]Given that the discriminant is negative, the square root term leads to imaginary values. Therefore, simplifying yields the complex solutions:\[ x = \frac{7 \pm i\sqrt{3}}{2} \]This formula confirms why understanding the discriminant first helps in anticipating the kind of solutions you can expect.