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Chapter 11: Problem 32

Explain why the expression \(b^{2}-4 a c\) from the quadratic formula will determine whether or not the solutions of a particular quadratic equation are imaginary.

Short Answer

Expert verified
A negative discriminant (\(b^2-4ac<0\)) results in imaginary solutions.

Step by step solution

01

Understanding the Quadratic Formula

The quadratic equation is given by \(ax^2 + bx + c = 0\), and the solutions for this equation are found using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
02

Identifying the Discriminant

The expression under the square root sign in the quadratic formula, \(b^2 - 4ac\), is known as the discriminant. It is crucial in determining the nature of the solutions.
03

Analyzing the Discriminant for Imaginary Solutions

The discriminant \(b^2 - 4ac\) tells us about the roots as follows: if \(b^2 - 4ac < 0\), the square root of a negative number results in an imaginary number. Thus, when the discriminant is negative, the quadratic equation has two complex (imaginary) solutions.

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Key Concepts

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

Discriminant
In the quadratic equation, the discriminant plays a vital role. It is the key expression found within the quadratic formula: \(b^2 - 4ac\). This component helps determine the type of solutions a quadratic equation has.
  • If the discriminant is positive \((b^2 - 4ac > 0)\), the quadratic equation has two distinct real solutions.
  • If it is zero \((b^2 - 4ac = 0)\), there is exactly one real solution, a special case known as a repeated or double root.
  • When the discriminant is negative \((b^2 - 4ac < 0)\), the solutions turn out to be imaginary. This means that the equation does not intersect the x-axis in a real coordinate system.

Understanding the discriminant allows students to predict the nature of the solutions even before solving the equation.
Imaginary Solutions
Imaginary solutions arise when you take the square root of a negative number. In the context of quadratic equations, this occurs when the discriminant \(b^2 - 4ac\) is less than zero.
  • This implies that the quadratic equation has no real solutions, meaning the parabola does not touch or cross the x-axis.
  • In practical terms, imaginary solutions involve a component labeled as \(i\), known as the imaginary unit.
  • The imaginary unit \(i\) is defined by the property that \(i^2 = -1\).

Imaginary solutions are complex in nature because they cannot be plotted on the real number line. Instead, they exist in a different dimension, often represented in the complex plane.
Complex Numbers
Complex numbers extend our number system by incorporating imaginary solutions. A complex number is composed of a real part and an imaginary part, usually written in the form: \(a + bi\).
  • Here, \(a\) is the real part, and \(bi\) is the imaginary part.
  • Complex numbers are essential tools in various fields, including electrical engineering and quantum physics.
  • They also facilitate operations and expressions that would be otherwise impossible with only real numbers.

When working with quadratic equations with a negative discriminant, the solutions can be expressed as complex numbers. For example, if the discriminant were -9, the solution might look like \(3i\), representing an imaginary solution.
Quadratic Equation Solutions
Solving quadratic equations entails finding the values of \(x\) that satisfy \(ax^2 + bx + c = 0\). The quadratic formula provides a universal method for this task: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
  • These solutions depend heavily on the discriminant \(b^2 - 4ac\).
  • Real solutions arise when the discriminant is non-negative, while imaginary solutions emerge when it is negative.
  • The solutions could also be expressed as a pair of complex numbers when dealing with imaginary solutions.

By analyzing the discriminant first, one can determine the nature of solutions for any given quadratic equation even before jumping into calculations.

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Most popular questions from this chapter

A pool service company adds neutralizer to pools on a monthly basis. The number of pounds of neutralizer used can be expressed as \(f(n)=0.0003 n\), where \(n\) is the number of gallons of water in the pool. Create a table showing the number of pounds of neutralizer used for pools with 10,000 gallons of water, 15,000 gallons of water, 20,000 gallons of water, 25,000 gallons of water, and 30,000 gallons of water.

Solve each of the following quadratic equations, and check your solutions. $$(5 x-1)^{2}=9$$

Solve each of the following quadratic equations, and check your solutions. $$(7 x+3)^{2}=1$$

The linear depreciation method assumes that an item depreciates the same amount each year. Suppose a new piece of machinery costs $$\$ 32,500$$, and it depreciates $$\$ 1950$$ each year for \(t\) years. Then we can express the value of the machinery after \(t\) years by the function \(V(t)=\) \(32,500-1950 t\). (a) Find the value of the machinery after 6 years. (b) Find the value of the machinery after 9 years. (c) Graph the linear function \(V(t)=32,500-1950 t\). (d) Use the graph from part (c) to approximate the value of the machinery after 10 years. Then use the function to find the exact value. (e) Use the graph to approximate how many years it will take for the value of the machinery to become zero. (f) Use the function to determine exactly how long it will take for the value of the machinery to become zero.

Solve each of the following quadratic equations, and check your solutions. $$(x+3)^{2}=-16$$
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