Problem 63 - Calculus: One and Several Variables

Question

Find the real roots of the equation. \(x^{2}+4 x+13=0\).

Step-by-Step Solution

Answer

The equation \(x^{2}+4 x+13=0\) does not have any real roots.

Step 1: Identify the coefficients

In the equation \(x^{2} + 4x + 13 = 0\), the coefficients corresponding to \(ax^{2} + bx + c = 0\) are a = 1, b = 4, and c = 13.

Step 2: Check the Discriminant

Calculate the discriminant \(D = b^{2} - 4ac\). Substituting the values, we get \(D = (4)^{2} - 4(1)(13) = 16 - 52 = -36\). Since the discriminant is less than 0, the roots of the equation are complex or imaginary, not real.

Step 3: Verify Results

If the result of the discriminant had been greater than 0, we would have computed the roots using the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\). However, since D < 0, we can confirm that no real roots exist for this equation.

Key Concepts

Discriminant

The discriminant is a crucial part of a quadratic equation in the form \(ax^2 + bx + c = 0\). It helps us determine the nature of the roots without actually solving the equation. You find the discriminant using the formula \(D = b^2 - 4ac\). Here is how it works:

• If \(D > 0\), there are two distinct real roots.
• If \(D = 0\), there is one real root, often called a repeated or double root.
• If \(D < 0\), there are no real roots, but instead, two complex or imaginary roots.

In the given equation \(x^{2} + 4x + 13 = 0\), when we calculate the discriminant \(D = 4^2 - 4 \times 1 \times 13 = -36\).
Since \(D < 0\), it tells us that the quadratic equation does not possess real roots. The roots must be complex, indicating the intersection with the x-axis does not occur.
Identifying the discriminant upfront can save time and aid in understanding the overall behavior of the quadratic equation.

Complex Roots

When the discriminant of a quadratic equation is less than zero, the roots are not real but complex. Complex roots arise from the part \(\sqrt{D} = \sqrt{-36}\) in our example. This expression involves the square root of a negative number, which leads us to imaginary numbers.
Complex roots always come in conjugate pairs, such as \(a + bi\) and \(a - bi\). In our equation \(x^2 + 4x + 13 = 0\), the roots can be explored using the quadratic formula:
\[x = \frac{-b \pm \sqrt{D}}{2a}\]
Substituting the values, we get:

• \(x = \frac{-4 \pm \sqrt{-36}}{2}\)

This simplifies to:

• \(x = \frac{-4 \pm 6i}{2}\)
• \(x = -2 \pm 3i\)

The \(i\) here represents the imaginary unit, defined by \(i^2 = -1\).
Understanding complex roots is important as they reflect scenarios and systems that exist beyond just the real number line, commonly found in advanced mathematics, physics, and engineering.

Coefficients

Coefficients in a quadratic equation, denoted typically as \(a\), \(b\), and \(c\) in the standard form \(ax^2 + bx + c = 0\), heavily influence the equation's attributes, particularly its roots.
Here's what each coefficient represents:

• **a**: The coefficient of the \(x^2\) term, it controls the parabola's opening direction. If \(a > 0\), it opens upwards, and if \(a < 0\), it opens downwards.
• **b**: The coefficient of the \(x\) term, it influences the vertex's horizontal position and the axis of symmetry.
• **c**: The constant term, it affects the vertical position of the parabola. Its value is where the parabola crosses the y-axis.

In the example \(x^2 + 4x + 13 = 0\), the coefficients are \(a = 1\), \(b = 4\), and \(c = 13\).
These numbers dictate not only the shape and position of the parabola but also how to calculate the discriminant and, consequently, determine the root nature. Coefficients interact seamlessly to refine the behavior and characteristics of quadratic equations across various applications. Keeping a close eye on them can offer valuable insights into problem-solving tasks involving quadratics.