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Chapter 0: Problem 69

Solve using any method. $$3 p^{2}-9 p+1=0$$

Short Answer

Expert verified
The solutions are \(p = \frac{9 + \sqrt{69}}{6}\) and \(p = \frac{9 - \sqrt{69}}{6}\).

Step by step solution

01

Identify the Coefficients

The given quadratic equation is \(3p^2 - 9p + 1 = 0\). Here, the coefficients are: \(a = 3\), \(b = -9\), and \(c = 1\).
02

Use the Quadratic Formula

The quadratic formula is \(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Apply this formula using the identified coefficients: \(a = 3\), \(b = -9\), and \(c = 1\).
03

Calculate the Discriminant

The discriminant is the part inside the square root of the quadratic formula: \(b^2 - 4ac\). Calculate it: \((-9)^2 - 4 \times 3 \times 1 = 81 - 12 = 69\).
04

Compute the Roots

Using the quadratic formula: \(p = \frac{-(-9) \pm \sqrt{69}}{2 \times 3}\). Simplify: \(p = \frac{9 \pm \sqrt{69}}{6}\). So, the roots are \(p = \frac{9 + \sqrt{69}}{6}\) and \(p = \frac{9 - \sqrt{69}}{6}\).

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Key Concepts

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

Coefficients
When working with quadratic equations, identifying the coefficients is a crucial first step. A quadratic equation is typically written in the form: \[ ax^2 + bx + c = 0 \]Here, the letters \(a\), \(b\), and \(c\) represent the coefficients of the terms in the equation. Each coefficient has a specific role:
  • \(a\): The coefficient of the squared term, \(x^2\).
  • \(b\): The coefficient of the linear term, \(x\).
  • \(c\): The constant term, which is a number without any variable attached.
In the equation \(3p^2 - 9p + 1 = 0\), we can see:
  • \(a = 3\)
  • \(b = -9\)
  • \(c = 1\)
Identifying these correctly is important for using other methods to solve the equation.
Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:\[p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula can always be used to find the roots of any quadratic equation, regardless of whether they are real or complex. The formula consists of different parts:
  • \(-b\): The opposite of the linear coefficient.
  • \(\pm\): Indicates that there are usually two solutions, one taking the plus and the other the minus in the equation.
  • Square Root: Involves the discriminant, \(b^2 - 4ac\), which we will discuss next.
  • \(2a\): Represents the denominator that multiplies the coefficient \(a\).
By substituting the coefficients into this formula, we obtain the roots of the quadratic equation neatly.
Discriminant
The discriminant is the part of the quadratic formula that lies under the square root, and it plays a significant role in determining the nature of the roots. It is computed as:\[b^2 - 4ac\]The value of the discriminant gives us insights into the roots:
  • If positive: The equation has two distinct real roots.
  • If zero: The equation has exactly one real root (a repeated root).
  • If negative: The equation has two complex roots.
In the example \(3p^2 - 9p + 1 = 0\), the discriminant is calculated as:\[(-9)^2 - 4 \times 3 \times 1 = 81 - 12 = 69\]Since 69 is positive, there are two distinct real roots for this equation.
Roots
The roots of a quadratic equation are the solutions for the variable \(p\) that satisfy the equation. Using the quadratic formula, the roots can be found by evaluating:\[p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]After calculating the discriminant and substituting it into the formula, the roots for the equation \(3p^2 - 9p + 1 = 0\) can be calculated as:\[p = \frac{9 \pm \sqrt{69}}{6}\]This computation results in two roots:
  • \(p = \frac{9 + \sqrt{69}}{6}\)
  • \(p = \frac{9 - \sqrt{69}}{6}\)
These roots represent the points where the quadratic equation intersects the horizontal axis when graphed, providing a complete solution to the equation.

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Most popular questions from this chapter

A phone provider offers two calling plans. Plan A has a \(\$ 30\) monthly charge and a \(\$ .10\) per minute charge on every call. Plan B has a \(\$ 50\) monthly charge and a \(\$ .03\) per minute charge on every call. Explain when Plan A is the better deal and when Plan \(\mathrm{B}\) is the better deal.

Determine whether the lines are parallel, perpendicular, or neither, and then graph both lines in the same viewing screen using a graphing utility to confirm your answer. $$\begin{aligned} &y_{1}=0,16 x+2.7\\\ &y_{2}=6.25 x-1.4 \end{aligned}$$

Home Improvement. The cost of having your bathroom remodeled is the combination of material costs and labor costs. The materials (tile, grout, toilet, fixtures, etc.) cost is \(\$ 1,200\) and the labor cost is \(\$ 25\) per hour. Write an equation that models the total cost \(C\) of having your bathroom remodeled as a function of hours \(h\). How much will the job cost if the worker estimates 32 hours?

Graph the function represented by each side of the equation in the same viewing rectangle and solve for \(x\) $$2 x+6=4 x-2 x+8-2$$

98\. Budget: Rental Car. The cost of a one-day car rental is the sum of the rental fee, \(\$ 50,\) plus \(\$ .39\) per mile. Write an equation that models the total cost associated with the car rental.
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