Problem 91 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, explain why or give an example to show why it is false. If \(b^{2}-4 a c>0\) and \(a \neq 0\), then the roots of \(a x^{2}-b x+\) \(c=0\) are the negatives of the roots of \(a x^{2}+b x+c=0 .\)

Step-by-Step Solution

Answer

The statement is false. A counterexample is provided by the quadratic equations \(x^2+1=0\) and \(x^2-1=0\), where the roots are not negatives of each other. More generally, the analysis of the roots shows that the roots of the given equations can only coincide when b=0, but this condition doesn't guarantee that the statement is true for all cases.

Step 1: Understand given conditions

In the given statement, we have a discriminant condition: \(b^2 - 4ac > 0\) and \(a \neq 0\). This implies the quadratic equations have real and distinct roots.

Step 2: Roots of the quadratic equation ax^2 - bx + c = 0

Let's find the roots of the equation \(ax^2 - bx + c = 0\). The roots can be found using the quadratic formula: \[x_{1,2} = \frac{-(-b) \pm \sqrt{(-b)^2 - 4ac}}{2a}\] \[x_{1,2} = \frac{b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 3: Roots of the quadratic equation ax^2 + bx + c = 0

Now let's find the roots of the equation \(ax^2 + bx + c = 0\). The roots can be found using the quadratic formula: \[x^*_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Step 4: Compare the roots of both equations

Now we need to compare the roots of the first equation with the negative roots of the second equation: \[x_{1,2} = -x^*_{1,2} \Leftrightarrow\] \[\frac{b \pm \sqrt{b^2 - 4ac}}{2a} = -\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] After simplifying the equation, we get: \[b \pm \sqrt{b^2 - 4ac} = -b \mp \sqrt{b^2 - 4ac}\] This equation can be true only if both the signs "+" and "–" match on either side. In that case: \[2b = 0 \Rightarrow b = 0\] However, having \(b=0\) doesn't guarantee that the statement is true for all cases; it only states it is a necessary condition.

Step 5: Draw a conclusion

Considering the analysis of the roots of the given quadratic equations, the statement is false. To show this, let's provide a counterexample where \(b^2-4ac > 0\), \(a\neq 0\), and \(b=0\): Consider the equations: \(x^2+1=0\) and \(x^2-1=0\). In this case, \(b=0\), and the roots are: For \(x^2+1=0\): \(x_{1,2} = \pm i\) For \(x^2-1=0\): \(x^*_{1,2} = \pm 1\) As we can see, the roots of the first equation are not the negatives of the roots of the second equation. Thus, the statement is false.

Key Concepts

Discriminant

The discriminant is a key component in the realm of quadratic equations, providing critical insights into the nature of their roots. The discriminant of a quadratic equation is found using the formula derived from the equation's coefficients: \[D = b^2 - 4ac\]
The value of the discriminant helps in determining the type of roots the quadratic equation will have:

• If \(D>0\), the equation has two real and distinct roots.
• If \(D=0\), the equation has exactly one real root, or put another way, it has two real, coincident roots.
• If \(D<0\), the equation has two complex roots.

Understanding the discriminant is critical, as it not only predicts the nature of the roots but also indicates whether the roots are rational or irrational. All this information is particularly useful when solving quadratic equations, allowing for a tailored approach depending on the characteristics of the discriminant value.

Quadratic Formula

The quadratic formula is a powerful tool that solves quadratic equations of the general form \(ax^2 + bx + c = 0\), where \(a eq 0\). It provides the solution for the roots of any quadratic equation directly and is formulated as follows:
\[x_{1,2} = \frac{-b \[pm\] \sqrt{b^2 - 4ac}}{2a}\]
It's a universally applicable method and can be broken down into several parts:

• \(-b\) represents the negation of the linear coefficient.
• The expression under the square root, \(b^2 - 4ac\), is the discriminant discussed earlier.
• The denominator \(2a\) scales the solution because the quadratic coefficient \(a\) influences the curvature of the parabola represented by the equation.

Using the quadratic formula simplifies the process of finding roots by eliminating the need to factor the quadratic expression, which can be complex or sometimes not possible when the roots are irrational or complex.

Real and Distinct Roots

When dealing with quadratic equations, the term 'real and distinct roots' refers to the solutions of the equation that are real numbers, and no two roots are the same. This situation arises when the discriminant of the quadratic equation is greater than zero, i.e., \(D > 0\).
For example, if we consider the quadratic equation \(ax^2 + bx + c = 0\), the condition for it to have real and distinct roots is manifest in the discriminant being positive. Here's how it works:

• If \(b^2 - 4ac > 0\), the square root of the discriminant, \(\sqrt{b^2 - 4ac}\), yields two different real numbers when added to and subtracted from \(-b\).
• This leads to the two distinct roots \(x_1\) and \(x_2\), derived from applying the '+' and '-' signs in the quadratic formula.

Hence, distinguishing between real and distinct roots is crucial as it signifies the presence of two unique intersections of the quadratic graph with the x-axis, reflecting the equation's distinct solutions.