Problem 60 - College Algebra and Calculus: An Applied Approach

Question

Solve the quadratic equation. $$x^{2}+6 x+10=0$$

Step-by-Step Solution

Answer

The roots of the quadratic equation \(x^2 + 6x + 10 = 0\) are -3+i and -3-i, which are complex numbers.

Step 1: Identifying coefficients

In the given equation \(x^2 + 6x + 10 = 0\), the coefficients are a=1, b=6, and c=10.

Step 2: Computing the discriminant

The discriminant can be calculated using the formula \(b^2 - 4ac\). So, our discriminant would be \( (6)^2 - 4*1*10 = 36 - 40 = -4\).

Step 3: Checking the discriminant

Since the discriminant (-4) is less than 0, this implies that the equation has complex (or imaginary) solutions.

Step 4: Solving for x

We solve for x using the formula \( x = [-b ± sqrt(b^2 - 4ac)]/2a \). Substituting the values from earlier gives \( x = [-6 ± sqrt(-4)]/2(1) = -3 ± sqrt(-1) = -3 ± i\).

Key Concepts

Complex Solutions

In math, when we talk about complex solutions, we're dealing with numbers that include the imaginary unit, commonly represented by the symbol \(i\). The imaginary unit \(i\) is defined as the square root of \(-1\). This is quite different from regular real numbers. When a quadratic equation has complex solutions, it means that the solutions are not on the real number line. Instead, they consist of a real part and an imaginary part.

For the quadratic equation \(x^2 + 6x + 10 = 0\), the solutions we found were \(-3 + i\) and \(-3 - i\). Here, \(-3\) is the real part, and \(\pm i\) are the imaginary parts. Complex solutions often appear in pairs known as complex conjugates, where the imaginary parts are equal and opposite, like \( a + bi \) and \( a - bi \). These solutions arise when the discriminant of a quadratic is negative, indicating no real roots.

Discriminant in Quadratic Equations

The discriminant is a crucial part of solving quadratic equations and understanding their nature. It's found using the formula \( b^2 - 4ac \), where \( a \), \( b \), and \( c \) are the coefficients from the standard quadratic equation \(ax^2 + bx + c = 0\). The value of the discriminant tells us a lot about the roots of the quadratic equation:

• If the discriminant is positive, there are two distinct real solutions.
• If the discriminant is zero, there is exactly one real solution (a repeated root).
• If the discriminant is negative, the solutions are complex or imaginary.

In our example, the equation \(x^2 + 6x + 10 = 0\) has a discriminant of \(-4\). This negative value reveals that we will have complex solutions, as we can see with \(-3 + i\) and \(-3 - i\). Remember, the discriminant is a shortcut that immediately lets you know what type of solutions to expect.

Solving Quadratic Equations

To solve quadratic equations, especially those that cannot be factored easily, you'll often use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \). This formula is derived from completing the square of a quadratic equation and allows you to find solutions for any quadratic equation.

Let's break down the process using our sample equation \(x^2 + 6x + 10 = 0\):

• First, determine the coefficients: \(a = 1\), \(b = 6\), and \(c = 10\).
• Calculate the discriminant: \(b^2 - 4ac = 36 - 40 = -4\).
• Since the discriminant is negative, expect complex solutions.
• Plug everything into the quadratic formula: \( x = \frac{-6 \pm \sqrt{-4}}{2(1)} = -3 \pm i \).

The quadratic formula is a reliable method that guarantees finding solutions regardless of whether they are real or complex. It liberates you from needing to factor, especially when dealing with more complicated polynomials.