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Chapter 0: Problem 105

Use the Quadratic Formula to solve the equation. $$ 9 x^{2}+24 x+16=0 $$

Short Answer

Expert verified
The roots of the equation \(9x^2 + 24x + 16 = 0\) is \(x=-4/3\)

Step by step solution

01

Identify the coefficients

In the given equation \(9x^2 + 24x + 16 = 0\), the coefficients are: a = 9, b = 24, c = 16.
02

Substitute the coefficients into the quadratic formula

Substitute a=9, b=24, and c=16 into the Quadratic Formula to get: \(x=\frac{-24\pm\sqrt{24^2-4*9*16}}{2*9}\)
03

Calculate the roots

The discriminant, \(b^2 - 4ac\), equals 0. So, the equation has exactly one root. Simplify the expression under the square root sign to find that \(x=\frac{-24\pm 0}{18}\). Thus, \(x=-24/18=-4/3\)

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Key Concepts

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

Quadratic Equation
A quadratic equation is a second-degree polynomial equation. This means it has the form:
  • \( ax^2 + bx + c = 0 \)
where \( a \), \( b \), and \( c \) are numbers called coefficients, and \( a eq 0 \) because otherwise it wouldn't be quadratic.
The special characteristic of a quadratic equation is that it can have at most two solutions, also known as roots. These roots can be real or complex, depending on the values of \( a \), \( b \), and \( c \).
To solve a quadratic equation, you can use several methods like factoring, completing the square, or using the Quadratic Formula. In this exercise, the Quadratic Formula is utilized to find the roots.
Coefficients
Coefficients in a quadratic equation are the numbers in front of the variables. In the equation \( 9x^2 + 24x + 16 = 0 \), the coefficients are:
  • \( a = 9 \) - this is the coefficient of \( x^2 \)
  • \( b = 24 \) - this is the coefficient of \( x \)
  • \( c = 16 \) - this is the constant term
These coefficients play a crucial role in determining the shape of the parabola and the position of its vertex on a graph.
They also determine the number and type of roots that the quadratic equation will have.
Without these coefficients, it would be impossible to apply the Quadratic Formula or any method to find the roots of the equation.
Discriminant
The discriminant is a key component of the Quadratic Formula. It is part of what determines the nature and number of roots in a quadratic equation. The discriminant is calculated using the formula:
  • \( b^2 - 4ac \)
In our exercise, the discriminant calculates to zero, where \( b = 24 \), \( a = 9 \), and \( c = 16 \).
  • If the discriminant is greater than zero, the quadratic equation has two distinct real roots.
  • If it is zero, as in our case, the equation has exactly one real root.
  • If the discriminant is less than zero, the roots are complex and involve imaginary numbers.
Thus, by determining the value of the discriminant, we get valuable information about the roots without having to calculate them entirely.
Roots
Roots of a quadratic equation are the solutions where the equation equals zero. They represent the \( x \)-values where the graph of the quadratic function touches or crosses the \( x \)-axis.
For the equation \( 9x^2 + 24x + 16 = 0 \), using the Quadratic Formula:
  • \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \)
Since the discriminant is zero for this exercise, we find just one root:
  • \( x = \frac{-24 \pm 0}{18} = -\frac{4}{3} \)
This root, \( x = -\frac{4}{3} \), means the parabola touches the \( x \)-axis at a single point, indicating it is a repeated root or double root.
This special circumstance shows the graph of the quadratic equation being tangent to the \( x \)-axis at one point rather than crossing it.

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Most popular questions from this chapter

(a) use the position equation \(s=-16 t^{2}+v_{0} t+s_{0}\) to write a function that represents the situation, (b) use a graphing utility to graph the function, (c) find the average rate of change of the function from \(t_{1}\) to \(t_{2}\), (d) interpret your answer to part (c) in the context of the problem, (e) find the equation of the secant line through \(t_{1}\) and \(t_{2}\), and (f) graph the secant line in the same viewing window as your position function. An object is thrown upward from ground level at a velocity of 96 feet per second. $$ t_{1}=2, t_{2}=5 $$

Transportation For groups of 80 or more people, a charter bus company determines the rate per person according to the formula Rate \(=8-0.05(n-80), \quad n \geq 80\) where the rate is given in dollars and \(n\) is the number of people. (a) Write the revenue \(R\) for the bus company as a function of \(n\). (b) Use the function in part (a) to complete the table. What can you conclude? \begin{tabular}{|l|l|l|l|l|l|l|l|} \hline\(n\) & 90 & 100 & 110 & 120 & 130 & 140 & 150 \\ \hline\(R(n)\) & & & & & & & \\ \hline \end{tabular}

Use a graphing utility to graph each function. Write a paragraph describing any similarities and differences you observe among the graphs. (a) \(y=x\) (b) \(y=x^{2}\) (c) \(y=x^{3}\) (d) \(y=x^{4}\) (e) \(y=x^{5}\) (f) \(y=x^{6}\)

College Students The numbers of foreign students \(F\) (in thousands) enrolled in colleges in the United States from 1992 to 2002 can be approximated by the model. $$ F=0.004 t^{4}+0.46 t^{2}+431.6, \quad 2 \leq t \leq 12 $$ where \(t\) represents the year, with \(t=2\) corresponding to 1992. (Source: Institute of International Education) (a) Use a graphing utility to graph the model. (b) Find the average rate of change of the model from 1992 to 2002. Interpret your answer in the context of the problem. (c) Find the five-year time periods when the rate of change was the greatest and the least.

Find the average rate of change of the function from \(x_{1}\) to \(x_{2}\). $$ \begin{array}{cc} \text { Function } & x \text {-Values } \\ \(f(x)=-\sqrt{x-2}+5 &\quad x_{1}=3, x_{2}=11\) \end{array} $$
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