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Chapter 4: Problem 57

Explain what is wrong with the statement. Every function of the form \(f(x)=x^{2}+b x+c,\) where \(b\) and \(c\) are constants, has two zeros.

Short Answer

Expert verified
The statement is incorrect as it doesn't account for cases where the discriminant \( b^2 - 4c \leq 0 \).

Step by step solution

01

Understanding Quadratic Functions

A quadratic function is generally defined as \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. In this case, \( a \) is not given explicitly, but it defaults to 1 since the leading term is \( x^2 \) alone.
02

Exploring the Concept of Zeros

The zeros of a function are the values of \( x \) for which \( f(x) = 0 \). For a quadratic function \( ax^2 + bx + c = 0 \), the zeros can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
03

Analyzing the Discriminant

The discriminant, given by \( \Delta = b^2 - 4ac \), determines the nature of the roots of a quadratic equation. If \( \Delta > 0 \), there are two distinct real zeros; if \( \Delta = 0 \), there is exactly one real zero (a repeated root); and if \( \Delta < 0 \), there are no real zeros but two complex zeros.
04

Applying the Discriminant to the Given Statement

For \( f(x)=x^2+bx+c \), the discriminant becomes \( b^2 - 4 \cdot 1 \cdot c = b^2 - 4c \). The statement claims every such function has two zeros, but this is only true if \( b^2 - 4c > 0 \). If \( b^2 - 4c \leq 0 \), there are not two distinct real zeros.
05

Identifying the Error in the Statement

The statement is incorrect because it assumes two zeros exist without considering the possibility of \( b^2 - 4c \leq 0 \). Quadratics with a discriminant \( b^2 - 4c < 0 \) have no real zeros, and those with \( b^2 - 4c = 0 \) have exactly one real zero.

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Key Concepts

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

Zeros of a Function
Zeros of a function are the points where the graph of the function touches or crosses the x-axis. For quadratic functions like \( f(x) = ax^2 + bx + c \), these points are critical because they help us understand the behavior of the function. To find these zeros, we set the function equal to zero and solve for \(x\):
  • For instance, for \( ax^2 + bx + c = 0 \), the values of \(x\) that satisfy this equation are the zeros of the function.
  • Using the quadratic formula, these values can be calculated as \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
This means the quadratic function can have zero, one, or two zeros depending on the nature of its roots.
It鈥檚 essential to note that zeros are sometimes also called roots or solutions of the equation \( f(x) = 0 \). Understanding zeros helps in graphing quadratic functions and solving real-world problems that boil down to quadratic equations.
Discriminant
The discriminant is a straightforward yet powerful component of the quadratic formula. It gives insights into the nature of the roots of the quadratic function, \( ax^2 + bx + c = 0 \). The discriminant is calculated by using \( \Delta = b^2 - 4ac \):
  • If \( \Delta > 0 \), the quadratic function has two distinct real zeros. This tells us the parabola crosses the x-axis at two points.
  • If \( \Delta = 0 \), there is exactly one real zero, meaning the parabola just touches the x-axis at one point, forming a "double root."
  • If \( \Delta < 0 \), there are no real zeros; instead, there are two complex zeros. The parabola does not intersect the x-axis at all in the real plane.
The discriminant thus helps determine the type and number of solutions without graphing the function. It provides a precise way to quickly judge the nature of the roots and link the algebraic equation to its geometric representation.
Real and Complex Roots
Roots of quadratic equations can be real or complex, highlighted by the value of the discriminant:
  • Real Roots: When \( \Delta \geq 0 \), the roots are real. For \( \Delta > 0 \), the roots are distinct and can be represented visually as points where the graph meets the x-axis. When \( \Delta = 0 \), the root is real but repeated (or coincidental), and it means the graph touches the x-axis at just one point.
  • Complex Roots: When \( \Delta < 0 \), the roots are complex and occur in conjugate pairs (\( p \pm qi \)). These roots don鈥檛 appear on the real number line. They reflect in cases where the graph of the quadratic does not intersect the x-axis.
Understanding these roots is crucial, especially in applications involving signals or systems that accommodate or rely on imaginary numbers. They show that quadratic equations can extend beyond simple real-number mathematics, offering solutions in broader mathematical contexts.

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Most popular questions from this chapter

In a \(19^{\text {th }}\) century sea-battle, the number of ships on each side remaining \(t\) hours after the start are given by \(x(t)\) and \(y(t) .\) If the ships are equally equipped, the relation between them is \((x(t))^{2}-(y(t))^{2}=c,\) where \(c\) is a positive constant. The battle ends when one side has no ships remaining. (a) If, at the start of the battle, 50 ships on one side oppose 40 ships on the other, what is the value of \(c ?\) (b) If \(y(3)=16,\) what is \(x(3) ?\) What does this represent in terms of the battle? (c) There is a time \(T\) when \(y(T)=0 .\) What does this \(T\) represent in terms of the battle? (d) At the end of the battle, how many ships remain on the victorious side? (e) At any time during the battle, the rate per hour at which \(y\) loses ships is directly proportional to the number of \(x\) ships, with constant of proportionality k. Write an equation that represents this, Is \(k\) positive or negative? (f) Show that the rate per hour at which \(x\) loses ships is directly proportional to the number of \(y\) ships, with constant of proportionality \(k\) (g) Three hours after the start of the battle, \(x\) is losing ships at the rate of 32 ships per hour. What is \(k ?\) At what rate is \(y\) losing ships at this time?

Graph the Lissajous figures using a calculator or computer. $$x=\cos 2 t, \quad y=\sin 4 t$$

Consider the family of functions \(y=f(x)=x-k \sqrt{x}\) with \(k\) a positive constant and \(x \geq 0 .\) Show that the graph of \(f(x)\) has a local minimum at a point whose \(x\) coordinate is \(1 / 4\) of the way between its \(x\) -intercepts.

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f\) is concave down and \(f(x)\) is positive for all \(x\).

Give an example of a function \(f\) that makes the statement true, or say why such an example is impossible. Assume that \(f^{\prime \prime}\) exists everywhere. \(f(x) f^{\prime \prime}(x)<0\) for all \(x\).
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