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Chapter 10: Problem 62

Solve equation. Approximate the solutions to the nearest hundredth when appropriate. \(\frac{1}{2} x^{2}+3 x+\frac{13}{2}=0\)

Short Answer

Expert verified
The solutions are complex roots: \(-3 + 2i\) and \(-3 - 2i\).

Step by step solution

01

Identify the Quadratic Form

The equation given is \( \frac{1}{2} x^{2}+3 x+\frac{13}{2}=0 \). This is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = \frac{1}{2} \), \( b = 3 \), and \( c = \frac{13}{2} \).
02

Use the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug the values of \( a \), \( b \), and \( c \) into this formula: \[ x = \frac{-3 \pm \sqrt{3^2 - 4 \times \frac{1}{2} \times \frac{13}{2}}}{2 \times \frac{1}{2}} \].
03

Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \[ 3^2 - 4 \times \frac{1}{2} \times \frac{13}{2} = 9 - 13 = -4 \]. The discriminant is \(-4\).
04

Identify the Nature of the Roots

Since the discriminant is negative, this implies that the equation has complex (non-real) roots. These roots can be expressed in the form involving \( i \), the imaginary unit.
05

Calculate the Complex Roots

Using the formula with \(-4\) under the square root: \[ x = \frac{-3 \pm \sqrt{-4}}{1} = -3 \pm i \sqrt{4} \]. Therefore, the roots are \( -3 + 2i \) and \( -3 - 2i \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula
The quadratic formula is a powerful tool used to find the roots of a quadratic equation. A quadratic equation is typically in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The quadratic formula itself is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula allows you to solve for \( x \), the variable of interest, by only substituting the values of \( a \), \( b \), and \( c \) from your quadratic equation.
  • \( -b \): Represents the negation of the coefficient of \( x \).
  • \( \sqrt{b^2 - 4ac} \): Known as the discriminant, helps determine the nature of the roots.
  • \( 2a \): Denominator that normalizes the result according to the leading coefficient \( a \).
Using this formula, one can compute both real and complex roots, making it universally applicable to any quadratic equation.
Complex Roots
Complex roots arise when a quadratic equation has a negative discriminant. In the quadratic formula, if the portion under the square root \((b^2 - 4ac)\) is negative, the result is imaginary. This is because the square root of a negative number is not real. To express complex roots, we use the imaginary unit \( i \), where \( i^2 = -1 \). For example, in our equation, the discriminant is \(-4\), indicating complex roots. Thus:
  • Substitute \(-4\) for \( b^2 - 4ac \) in the quadratic formula.
  • Compute the square root of the negative discriminant: \( \sqrt{-4} = 2i \).
  • The roots are then expressed as \( -3 \pm 2i \), reflecting both the real part \(-3\) and the imaginary part \( \pm 2i \).
Understanding complex roots is essential as they reflect the full range of solutions, even when the solutions do not cross the x-axis in a conventional, real-number sense.
Discriminant
The discriminant is a key component of the quadratic formula, found in the expression \( b^2 - 4ac \). It provides insight into the nature of the roots of a quadratic equation:
  • If \( b^2 - 4ac > 0 \), the quadratic equation has two distinct real roots.
  • If \( b^2 - 4ac = 0 \), there's exactly one real root (also known as a repeated or double root).
  • If \( b^2 - 4ac < 0 \), the roots are complex, meaning they include the imaginary unit \( i \).
In our equation, the discriminant is \(-4\). This negative value leads us to conclude that the roots are complex numbers, specifically \(-3 \pm 2i\). Understanding the discriminant helps to predict not just the quantity of roots, but their nature as well, offering a comprehensive view of the solution landscape.

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