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Chapter 2: Problem 85

In Exercises 83 - 86, (a) find the interval(s) for \( b \) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. \( 3x^2 + bx + 10 = 0 \)

Short Answer

Expert verified
The interval for b such that the equation has at least one real solution is (-infinity, -sqrt(120)] union [sqrt(120), +infinity).

Step by step solution

01

Understand the Given Equation

The given quadratic equation is \(3x^2 + bx + 10 = 0\). Here a = 3, b is the variable we need to find intervals for and c = 10.
02

Set Up Discriminant Condition for at Least One Real Root

For at least one real solution, our condition is Discriminant \(b^2 - 4ac\) should be greater than or equal to 0. Substituting a = 3 and c = 10, we get \(b^2 - 4*3*10 >= 0\) which simplifies to \(b^2>= 120\).
03

Find Values of b

To satisfy \(b^2>= 120\), b must be either greater or equal to \(\sqrt{120}\) or less or equal to \(-\sqrt{120}\). This gives us b belongs to \(-\infty, -\sqrt{120}\] or \([\sqrt{120}, +\infty\).
04

Write Conjecture Based on Coefficients

The possible intervals for b are entirely based on values of coefficients a and c in the given equation. If a, b, c were different, this would lead to different intervals for b.

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Key Concepts

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

Discriminant
When solving quadratic equations, the discriminant plays a crucial role in determining the nature and number of solutions. For a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is represented as \( b^2 - 4ac \).
The value of the discriminant helps us understand whether the roots of the quadratic equation are real or imaginary. Here's what different discriminant values indicate:
  • If the discriminant ( \(b^2 - 4ac\) ) is greater than 0, the quadratic equation has two distinct real solutions.
  • If the discriminant equals 0, there is exactly one real solution, also known as a repeated or double root.
  • If the discriminant is less than 0, then the equation has no real solutions. Instead, it has two complex conjugate solutions.
In our exercise, the condition \( b^2 - 4ac \geq 0 \) ensures that the quadratic has at least one real solution. This condition allows us to determine intervals for the variable \( b \).
Real Solutions
Real solutions refer to the values of \( x \) that satisfy the quadratic equation and are real numbers, as opposed to complex or imaginary numbers. The decision of when an equation has real solutions is directly related to its discriminant value.
  • For example, in the quadratic equation \( 3x^2 + bx + 10 = 0 \), we set the discriminant \( b^2 - 4ac \geq 0 \) to guarantee the existence of real solutions.
  • This means finding those values of \( b \) such that when substituted, the discriminant equation does not turn negative, thus providing at least one real value for \( x \).
So, the real solutions manifest when \( b^2 \geq 120 \), allowing us to identify intervals for \( b \) that produce realistic and verifiable outcomes in the equation's solution.
Coefficients
Coefficients in a quadratic equation, \( ax^2 + bx + c = 0 \), are the numerical factors \( a \), \( b \), and \( c \). Each of these coefficients plays a distinct role in shaping the quadratic equation's graph and determining its solutions.
For our equation \( 3x^2 + bx + 10 = 0 \):
  • \( a = 3 \) is the coefficient of the \( x^2 \) term and affects the parabola's width and direction (open upwards since \( a > 0 \)).
  • \( b \) is variable in this problem, impacting the orientation and axis of symmetry of the parabola.
  • \( c = 10 \) is the constant term which affects the vertical translation of the graph.
By altering the coefficient \( b \), we discern which intervals result in at least one real solution by maintaining the balance \( b^2 - 4*3*10 \geq 0 \). This teaches us how sensitive the quadratic nature is concerning the coefficients and, specifically, how \( b \) alters discriminant calculations.
Interval Notation
Interval notation provides a simple way of expressing ranges of numbers, often used when discussing solutions to inequalities. It clarifies the set of values a variable can take. In this context, after solving \( b^2 \geq 120 \), interval notation was used to express the permissible values of \( b \).
When inequalities result in strings like \(-\sqrt{120} \leq b \) or \( b \geq \sqrt{120} \), we can write:
  • Interval \( (-\infty, -\sqrt{120}] \) indicates \( b \) can take any value less than or equal to \(-\sqrt{120} \).
  • Interval \( [\sqrt{120}, +\infty) \) implies \( b \) can be any value greater than or equal to \(\sqrt{120} \).
By using interval notation, we can concisely and clearly specify the valid intervals for \( b \). It becomes a valuable format, especially for expressing solutions derived from continuous sets of numbers in mathematics.

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