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Chapter 7: Problem 37

Without solving the given equations, determine the character of the roots. $$3.6 t^{2}+2.1=7.7 t$$

Short Answer

Expert verified
The equation has two distinct real roots.

Step by step solution

01

Write the equation in standard form

The given equation is \(3.6t^2 + 2.1 = 7.7t\). Begin by rewriting this equation in the standard quadratic form \(at^2 + bt + c = 0\). Subtract \(7.7t\) from both sides to achieve this: \[3.6t^2 - 7.7t + 2.1 = 0\]
02

Identify coefficients

In the standard form of the quadratic equation \(at^2 + bt + c = 0\), identify the coefficients: \(a = 3.6\), \(b = -7.7\), and \(c = 2.1\).
03

Calculate the discriminant

The character of the roots of a quadratic equation is determined by its discriminant, \(\Delta = b^2 - 4ac\). Substitute the values of the coefficients: \[\Delta = (-7.7)^2 - 4 \times 3.6 \times 2.1\] Calculate the values: \[\Delta = 59.29 - 30.24\] \[\Delta = 29.05\]
04

Determine the nature of the roots

The discriminant \(\Delta\) determines the nature of the roots: - If \(\Delta > 0\), the quadratic equation has two distinct real roots.- If \(\Delta = 0\), the quadratic equation has exactly one real root (a repeated root).- If \(\Delta < 0\), the quadratic equation has two complex conjugate roots.Since \(\Delta = 29.05 > 0\), the quadratic equation has two distinct real roots.

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Key Concepts

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

Discriminant
The discriminant is a fundamental concept in determining the characteristics of the roots of a quadratic equation. When you have a quadratic equation in standard form, which is expressed as \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using the formula:
  • \( \Delta = b^2 - 4ac \)
The purpose of the discriminant is to identify the nature of the roots without actually solving the equation. It's like a shortcut to understand more about the roots. The discriminant can indicate three different scenarios:
  • If \( \Delta > 0 \), the equation has two distinct real roots.
  • If \( \Delta = 0 \), there is exactly one real root, known as a double or repeated root.
  • If \( \Delta < 0 \), the equation has two complex roots, which are not real numbers.
By recognizing these outcomes, we can learn about the solution set of the equation from the discriminant alone.
Real Roots
Real roots are the solutions to a quadratic equation that lie within the set of real numbers. In simpler terms, they're the x-values at which the graph of the quadratic equation touches or crosses the x-axis. These are important in many mathematical contexts because they are the solutions that you can "see" or have practical interpretations in real-world problems. When analyzing the roots of a quadratic equation, knowing the value of the discriminant is essential because:
  • When \( \Delta > 0 \), there are two distinct points of intersection with the x-axis, giving us two unique real roots.
  • If \( \Delta = 0 \), the parabola touches the x-axis at only one point, resulting in a single real root that is repeated.
  • If \( \Delta < 0 \), the graph does not intersect the x-axis, and there are no real roots, only complex ones.
This understanding of real roots aids in predicting the number of solutions and the behavior of graphs in algebraically complex scenarios.
Standard Form
The standard form of a quadratic equation is instrumental because it organizes the equation in a way that easily reveals its coefficients, crucial for further analysis. The standard form of a quadratic equation is expressed as:
  • \( ax^2 + bx + c = 0 \)
In this structure:
  • \( a \) is the coefficient of the squared term, \( x^2 \).
  • \( b \) is the coefficient of the linear term, \( x \).
  • \( c \) is the constant term, independent of \( x \).
Rewriting any given quadratic equation into this form is often step one when solving or analyzing the equation. It allows for straightforward calculation of the discriminant and facilitates the entire process of determining the nature of the roots. This standardized form is a universal language in mathematics that reveals both the potential behavior of the function and its solutions.

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Most popular questions from this chapter

Solve the given problems. All numbers are accurate to at least two significant digits. In machine design, in finding the outside diameter \(D_{0}\) of a hollow shaft, the equation \(D_{0}^{2}-D D_{0}-0.25 D^{2}=0\) is used. Solve for \(D_{0}\) if \(D=3.625 \mathrm{cm}\).

Use a calculator to graph all three parabolas on the same coordinate system. . Describe (a) the shifts and (b) the stretching and shrinking. (a) \(y=x^{2}\) (b) \(y=3 x^{2}\) (c) \(y=\frac{1}{3} x^{2}\)

Solve the given applied problem. Find the range of the function \(s=-16 t^{2}+64 t+6\).

Use a calculator to solve the given equations. Round solutions to the nearest hundredth. If there are no real roots, state this. $$3 x^{2}-25=20 x$$

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$20 r^{2}=20 r+1$$
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