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Chapter 3: Problem 29

Solve each equation. For equations with real solutions, support your answers graphically. $$x(14 x+1)=3$$

Short Answer

Expert verified
The solutions are \( x = \frac{3}{7} \) and \( x = -\frac{1}{2} \).

Step by step solution

01

Expand the Equation

Expand the given equation \( x(14x + 1) = 3 \) to a standard quadratic form. Distribute \( x \) inside the parentheses to get \( 14x^2 + x = 3 \).
02

Rearrange into Standard Form

Rearrange the expanded equation into the standard quadratic form \( ax^2 + bx + c = 0 \). Move 3 to the left side to get \( 14x^2 + x - 3 = 0 \).
03

Identify Coefficients

Identify the coefficients from the standard form equation \( 14x^2 + x - 3 = 0 \). Here, \( a = 14 \), \( b = 1 \), and \( c = -3 \).
04

Apply the Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions. Substitute the coefficients into the formula: \( x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 14 \cdot (-3)}}{2 \cdot 14} \).
05

Simplify the Discriminant

Calculate the discriminant \( b^2 - 4ac \). For this equation, \( 1 - 4 \cdot 14 \cdot (-3) = 1 + 168 = 169 \).
06

Calculate the Square Root

Find the square root of the discriminant: \( \sqrt{169} = 13 \).
07

Solve for x

Substitute the square root back into the quadratic formula: \( x = \frac{-1 \pm 13}{28} \). This gives two possibilities: \( x = \frac{-1 + 13}{28} = \frac{12}{28} = \frac{3}{7} \) and \( x = \frac{-1 - 13}{28} = \frac{-14}{28} = -\frac{1}{2} \).
08

Graphical Verification

To graphically verify, plot the quadratic equation \( y = 14x^2 + x - 3 \) on a coordinate plane. Notice that the curve intersects the x-axis at \( x = \frac{3}{7} \) and \( x = -\frac{1}{2} \), confirming the solutions.

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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 that helps find solutions to quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). Quadratic equations have coefficients \( a \), \( b \), and \( c \) that determine their shape and position when graphed.
  • The formula itself is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • It provides solutions by calculating the points where the parabola (graph) intersects the x-axis, known as the roots.
  • The \( \pm \) symbol means there can be two solutions: one with addition and one with subtraction.
This formula essentially "solves" for \( x \), delivering two possible values. For instance, in our exercise, it provided solutions of \( x = \frac{3}{7} \) and \( x = -\frac{1}{2} \). Using the Quadratic Formula makes solving complex equations much simpler.
Discriminant
The Discriminant is an important part of the Quadratic Formula. It is represented by \( b^2 - 4ac \) and helps you determine the nature of the roots of the quadratic equation.
  • If the Discriminant is positive, there are two distinct real roots.
  • If it is zero, it means there is one real root, also known as a repeated or double root.
  • If the Discriminant is negative, then there are no real roots, but two complex roots instead.
In our case, the Discriminant was \( 169 \), which is positive. This confirmed the existence of two distinct real solutions: \( x = \frac{3}{7} \) and \( x = -\frac{1}{2} \). Calculating the Discriminant can give valuable insights into the solutions without needing to solve the entire equation.
Graphical Solution
A Graphical Solution involves plotting the quadratic equation on a graph to visually identify the roots. Here, we graph the equation \( y = 14x^2 + x - 3 \) as a parabola.
  • The x-intercepts, where the curve meets the x-axis, indicate the real solutions of the equation.
  • In our example, these points were \( x = \frac{3}{7} \) and \( x = -\frac{1}{2} \), exactly matching the results from the Quadratic Formula.
  • Graphing helps in understanding the concept of roots visually, making abstract math more tangible.
Even if you have solved the equation algebraically, plotting the graph can reinforce your understanding and verify the accuracy of your solutions.
Standard Form of Quadratic
The Standard Form of a Quadratic Equation is \( ax^2 + bx + c = 0 \). Converting equations into this form makes them easier to analyze and solve using formulas and graphical methods.
  • "a" represents the coefficient of \( x^2 \), responsible for the parabola's "width" and direction (upward or downward).
  • "b" is the linear coefficient influencing the tilt and orientation of the parabola.
  • "c" is the constant term that determines the y-intercept of the graph.
In our exercise, the original equation \( x(14x + 1) = 3 \) was transformed into \( 14x^2 + x - 3 = 0 \), aligning it into the standard form. This rearrangement is crucial as it prepares the equation for applying the Quadratic Formula and aids in accurately identifying its characteristics and solutions.

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Most popular questions from this chapter

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$2 x^{2}-4 x+1=0$$

Evaluate the discriminant, and use it to determine the number of real solutions of the equation. If the equation does have real solutions, tell whether they are rational or irrational. Do not actually solve the equation. $$3 x^{2}=4 x-5$$

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}+5 x<2\) (b) \(x^{2}+5 x \geq 2\)

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+5)(x-6)=(2 x-1)(x-4)$$

Solve each problem. As a function of age group \(x,\) the fatality rate (per \(100,000\) population) for males killed in automobile accidents can be approximated by $$f(x)=1.8 x^{2}-12 x+37.4$$ where \(x=0\) represents ages \(21-24, x=1\) represents ages \(25-34, x=2\) represents ages \(35-44,\) and so on. Find the age group at which the accident rate is a minimum, and find the minimum rate. (Source: National Highway Traffic Safety Administration.)
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