Complete problems 48 and 67 Complete problem 89Show all o…
Complete problems 48 and 67 Complete problem 89 Show all of your work. You may type in Word using Equation Editor or write neatly by hand and upload as a PDF. Include copies of any graphs you make, whether by hand or using graphing software.
Answer
To solve the given problems, we will employ various mathematical techniques and principles. Problem 48 involves determining the roots of a quadratic equation, problem 67 deals with solving systems of linear equations, and problem 89 requires graphing a given function.
Let’s start by addressing problem 48. Quadratic equations are second-degree polynomial equations in the form of ax^2 + bx + c = 0, where a, b, and c are constants. To find the roots of a quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 – 4ac)) / (2a)
By substituting the given values of a, b, and c into this formula, we can calculate the roots. Ensure that all numerical values and operations are accurate while performing the calculations.
Moving on to problem 67, we are tasked with solving a system of linear equations. A system of linear equations consists of two or more linear equations with the same variables. We aim to find values for these variables that satisfy all the given equations simultaneously.
One commonly used method for solving systems of linear equations is substitution. We choose one equation and solve it for one variable in terms of the others. Then, we substitute this expression into the remaining equations to obtain a simplified system, which we can further solve to find the values of the variables.
Another method is called the elimination method. In this approach, we transform the system of equations so that when added or subtracted vertically, one variable is eliminated while solving for the other variable. This process is repeated until only one variable remains, allowing us to calculate its value. We then substitute this value back into one of the equations to find the value of the remaining variable.
Lastly, we come to problem 89, which involves graphing a given function. Graphing functions helps us visualize the behavior of the function and gain insights into its properties. When graphing a function, it is important to understand the key features and characteristics, such as intercepts, asymptotes, and the overall shape of the curve.
To graph the given function, we must plot points on a coordinate plane by substituting different values of x into the function and calculating the corresponding y-values. By connecting these points, we can create a graph of the function.
It is also crucial to label the axes, include appropriate scales, and title the graph with the function’s name or equation. Additionally, any significant points or regions should be highlighted, and graphing software can be employed to create accurate and visually appealing graphs.
To complete these problems, it is advisable to use mathematical software or tools such as equation editors, graphing software, or PDF creators to ensure neatness and accuracy when presenting the solutions and accompanying graphs.