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Chapter 16: Problem 16

Solve by using the quadratic formula. $$3 r^{2}-r+2=0$$

Short Answer

Expert verified
The roots of the quadratic equation \(3r^2 - r + 2 = 0\) are \(r = \frac{1}{6} \pm \frac{\sqrt{23}}{6}i\), where 'i' denotes the imaginary unit.

Step by step solution

01

Identify a, b, and c

In the given quadratic equation, \(3r^2 - r + 2 = 0\), the values of 'a', 'b', and 'c' can be identified as follows: 'a' is 3, 'b' is -1, and 'c' is 2.
02

Calculate the discriminant

The discriminant (indicated by the Greek letter Delta), given by \(b^2 - 4ac\), helps to determine the number and type of roots of the quadratic equation. For \(-1^2 - 4 * 3 * 2\), it calculates to -23.
03

Apply the Quadratic Formula

Use the quadratic equation to find the roots. Substituting the values 'a' = 3, 'b' = -1 and discriminant = -23, we calculate the solutions as \(r = \frac{-(-1) \pm \sqrt{-23}}{2*3}\) or \(r = \frac{1 \pm \sqrt{-23}}{6}\) . Since the discriminant is negative, the roots are complex and can be written in the form \(a + bi\). This gives the solutions \(r = \frac{1}{6} \pm \frac{\sqrt{23}}{6}i\).

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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 one of the most effective methods for solving quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). This powerful tool allows you to find the "roots" or solutions of any quadratic equation. The formula is presented as:
  • \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
To use the quadratic formula, it's fundamental to identify the coefficients \(a\), \(b\), and \(c\) from the equation. In the expression \(3r^2 - r + 2 = 0\), \(a\) is 3, \(b\) is -1, and \(c\) is 2. By substituting these into the formula, you'll be ready to find the solutions of the quadratic equation.
Discriminant
The discriminant is a key component of the quadratic formula and plays a crucial role in determining the nature of the roots of a quadratic equation. It is given by the expression \(b^2 - 4ac\). The value of the discriminant provides insights into the types of roots you can expect:
  • If the discriminant is positive, there are two distinct real roots.
  • If the discriminant is zero, there is one real root (a repeated root).
  • If the discriminant is negative, the equation has complex roots.
For the equation \(3r^2 - r + 2 = 0\), calculating the discriminant yields \((-1)^2 - 4 \times 3 \times 2 = -23\). Since the discriminant is negative, this equation has complex roots.
Complex Roots
When a quadratic equation has a negative discriminant, it means the solutions are complex numbers rather than real numbers. Complex roots occur in conjugate pairs and are of the form \(a + bi\) and \(a - bi\), where \(i\) represents the imaginary unit (\(i^2 = -1\)).
Implementing this to the equation \(3r^2 - r + 2 = 0\), with a negative discriminant \(-23\), we found that the roots are:
  • \(r = \frac{1}{6} + \frac{\sqrt{23}}{6}i\)
  • \(r = \frac{1}{6} - \frac{\sqrt{23}}{6}i\)
These roots reflect that the quadratic equation doesn't intersect the real number line, highlighting the importance of complex numbers in providing solutions to all polynomial equations.
Solving Equations
Solving quadratic equations often involves identifying the correct method based on the equation's characteristics. The quadratic formula is universally applicable, providing solutions for equations where factoring or completing the square might be difficult or impossible.
In the case of the equation \(3r^2 - r + 2 = 0\), following these steps:
  • Identify \(a, b,\) and \(c\) and compute the discriminant.
  • Substitute these into the quadratic formula.
  • Solve the formula while taking into account complex numbers if the discriminant is negative.
Always remember, the quadratic formula will yield either real or complex roots, ensuring every quadratic equation is solvable within the complex number system.

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