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Chapter 1: Problem 110

First determine whether the solutions of each quadratic equation are real or complex without solving the equation. Then solve the equation. \(-3 x^{2}-x-4=0\)

Short Answer

Expert verified
The solutions are complex: \(x = -\frac{1}{6} \pm \frac{i\sqrt{47}}{6}\).

Step by step solution

01

Determine the Discriminant

To find out if the solutions are real or complex, we calculate the discriminant of the quadratic equation. The formula for the discriminant is given by \(\Delta = b^2 - 4ac\), where \(a = -3\), \(b = -1\), and \(c = -4\) in our equation. Substitute these values to get \(\Delta = (-1)^2 - 4(-3)(-4)\).
02

Calculate the Discriminant

Substitute the values into the discriminant formula: \(\Delta = 1 - 48 = -47\). Since the discriminant is less than zero (\(\Delta < 0\)), the solutions are complex numbers.
03

Use the Quadratic Formula

Since the solutions are complex, we will use the quadratic formula: \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substitute \(a = -3\), \(b = -1\), and \(\Delta = -47\) into the formula to get \(x = \frac{-(-1) \pm \sqrt{-47}}{2(-3)}\).
04

Simplify the Quadratic Formula

Simplify the expression: \(x = \frac{1 \pm \sqrt{-47}}{-6}\). Since \(\sqrt{-47} = i\sqrt{47}\), further simplify it to \(x = \frac{1 \pm i\sqrt{47}}{-6}\).
05

Write the Final Solution

The solutions in terms of real and imaginary parts are \(x = -\frac{1}{6} \pm \frac{i\sqrt{47}}{6}\). Therefore, the solutions are complex numbers.

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Key Concepts

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

Discriminant
When you encounter a quadratic equation, the discriminant helps you determine the nature of its solutions. The discriminant is part of the quadratic formula, represented by \( b^2 - 4ac \).
  • If the discriminant is positive (\( \Delta > 0 \)), the quadratic equation has two distinct real solutions.
  • If the discriminant is zero (\( \Delta = 0 \)), the equation has exactly one real solution, also called a repeated or double root.
  • If the discriminant is negative (\( \Delta < 0 \)), the solutions are not real numbers; they are complex numbers.
In our original exercise, the discriminant was \(-47\), a negative number. This indicates the solutions to the quadratic equation are complex, not crossing the x-axis on the graph.
Complex Numbers
Complex numbers are numbers that have both real and imaginary parts. They are typically written in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) being the imaginary unit.
  • The imaginary unit \( i \) is defined as the square root of -1, so \( i^2 = -1 \).
  • Complex numbers can be added, subtracted, multiplied, and divided like real numbers, accounting for the imaginary part using \( i \).
In the solved equation, we ended up with solutions in terms of complex numbers: \( x = -\frac{1}{6} \pm \frac{i\sqrt{47}}{6} \). This means each solution has a real part \(-\frac{1}{6}\) and an imaginary part \( \pm \frac{\sqrt{47}}{6} i\). Understanding complex numbers is crucial, especially when dealing with negative discriminants.
Quadratic Formula
The quadratic formula is a universal tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). It is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula lets us find the solutions (roots) of the equation by substituting the coefficients \( a \), \( b \), and \( c \).
  • The symbols \( \pm \) suggest there will be two solutions, one with a plus and the other with a minus.
  • The square root of the discriminant \( \sqrt{b^2 - 4ac} \) determines if the solutions are real or complex.
  • For complex solutions, expressed using an imaginary unit \( i \), the process leads to results outside the typical real numbers.
For our exercise, using \( a = -3 \), \( b = -1 \), and a negative discriminant \(-47\), the quadratic formula results in solutions that are expressed in terms of complex numbers. This shows the flexibility of the quadratic formula, even when working with imaginary numbers.

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Most popular questions from this chapter

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the \mathrm{\\{} r e s u l t i n g ~ l i n e a r ~ r e l a t i o n s h i p ~ o n ~ a ~ l o g - l o g ~ p l o t . ~ $$ y=6 x^{-1} $$

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ N(t)=130 \times 2^{1.2 t} $$

Drug Absorption After a patient takes the painkiller acetaminophen (often sold under the brand name Tylenol), the concentration of drug in their blood increases at first, as the painkiller is absorbed into the blood, and then starts to decrease as the drug is metabolized or removed by the liver. In one study, the concentration of drug \((c\), measured in \(\mu \mathrm{g} / \mathrm{ml}\) ) was measured in a patient as a function of time \((t\), measured in hours since the drug was administered). The data in this example is taken from Rowling et al. (1977). \begin{tabular}{lc} \hline \(\boldsymbol{t}\) & \multicolumn{1}{c} {\(\boldsymbol{c}\)} \\ \hline 1 & \(10.61\) \\ \(1.5\) & \(8.73\) \\ 2 & \(7.63\) \\ 3 & \(5.55\) \\ 4 & \(3.97\) \\ 5 & \(3.01\) \\ 6 & \(2.39\) \\ \hline \end{tabular} (a) You want to determine from the data whether the relationship between concentration and time follows a power law $$ c=a t^{b} $$ for some set of constants \(a\) and \(b\), or whether it instead follows an exponential law $$ c=k d^{t} $$ for some constants \(k\) and \(d\). Explain how you could plot the data with transformed horizontal and vertical axes to determine which mathematical model is correct. (b) By plotting \(\log c\) against \(\log t\) in one graph, and \(\log c\) against \(t\) in another, explain why the data supports the second model (exponential decay) better than it supports the first model. (c) From your plot of \(\log c\) against \(t\), estimate the parameter \(d\).

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ R(t)=3.6 t^{1.2} $$

This problem relates to self-thinning of plants (Example 8). If we plot the logarithm of the over age weight of a plant, \(\log w\), against the logarithm of the density of plants, \(\log d\) (base 10 ), a straight line with slope \(-3 / 2\) results. Find the equation that relates \(w\) and \(d\), assuming that \(w=1 \mathrm{~g}\) when \(d=10^{3} \mathrm{~m}^{-2}\).
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