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Chapter 2: Problem 43

\(43-50\) . Solve the equation graphically in the given interval. State each answer rounded to two decimals. $$ x^{2}-7 x+12=0 ; \quad[0,6] $$

Short Answer

Expert verified
The solutions are \(x = 3.00\) and \(x = 4.00\).

Step by step solution

01

Understand the Equation

The given equation is a quadratic equation of the form \(x^2 - 7x + 12 = 0\). Our task is to find the solutions graphically within the interval \([0, 6]\).
02

Graph the Quadratic Function

Graph the quadratic function \(f(x) = x^2 - 7x + 12\). This parabola opens upwards since the coefficient of \(x^2\) is positive. We need to find where this graph intersects the x-axis in the specified interval \([0, 6]\).
03

Identify the x-axis Intersections

By plotting the function, observe where the curve crosses the x-axis. The x-values at these intersection points are the solutions to the equation \(x^2 - 7x + 12 = 0\).
04

Find Exact Solutions

Using the common values where the graph intersects the x-axis, identify that the roots of the quadratic within the interval \([0, 6]\) occur at \(x = 3\) and \(x = 4\).
05

Round the Answers

Round the solutions to two decimal places. Since \(x = 3\) and \(x = 4\) are already integers, they remain the same when rounded to two decimal places.

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Key Concepts

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

Graphical Solutions
Graphical solutions are a powerful way to solve quadratic equations. Instead of relying purely on algebraic methods, you can visualize the problem and gain insights from it.

To solve the equation by graphing, start by plotting the quadratic function on a coordinate plane. For example, consider the function given by the equation \(f(x) = x^2 - 7x + 12\).
  • The graphical solution involves finding where the parabola intersects the x-axis.
  • The x-intercepts represent the values that satisfy the equation \(x^2 - 7x + 12 = 0\).
Once plotted, these intersection points, or roots, are visually evident, providing an intuitive method to understand and solve the equation. This is especially useful when dealing with more complex functions or when looking for approximate solutions.
Parabola
A parabola is a specific type of graph that represents a quadratic function. It's a U-shaped curve that can open upwards or downwards.
In the equation \(x^2 - 7x + 12\), the parabola opens upwards since the coefficient of \(x^2\) is positive. This is an important characteristic, as it helps determine the nature of the roots.
  • The vertex, or the lowest point of this parabola, can provide insights into the function's behavior.
  • Understanding whether a parabola opens upwards or downwards helps predict whether it has real roots and how many there might be.
Recognizing the shape of the parabola is crucial when solving quadratics graphically, as it makes locating the roots through intersection points easier.
Roots of Quadratic
The roots of a quadratic equation are the x-values where the graph of the quadratic intersects the x-axis. In mathematical terms, these are the solutions to the equation. For the function \(x^2 - 7x + 12\), these roots occur at the points where the parabola intersects the x-axis.

Finding the exact roots:
  • On the graph of \(x^2 - 7x + 12\), observe the intersection points to find that the roots are at \(x = 3\) and \(x = 4\).
  • The roots are the values that make the quadratic equation equal zero.
By knowing how to find roots graphically, one gains a deeper understanding of how the algebraic and geometric representations of a quadratic equation relate to each other.
Interval Notation
Interval notation is a concise way to describe the set of values that satisfy certain conditions. It's often used to specify the range of solutions or to define the domain and range for which the solutions are valid.

In this problem, interval notation specifies the range \([0, 6]\). This means we're only interested in the parts of the graph within these x-values.
  • An interval like \([0, 6]\) indicates all x-values from 0 to 6, including the endpoints.
  • This lets us focus on just a portion of the graph, ignoring any irrelevant solutions that lie outside the specified interval.
Understanding interval notation helps narrow down the graph's domain to relevant solutions, making the solving process focused and efficient.

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Most popular questions from this chapter

Skidding in a Curve A car is traveling on a curve that forms a circular arc. The force \(F\) needed to keep the car from skidding is jointly proportional to the weight \(w\) of the car and the square of its speed \(s,\) and is inversely proportional to the radius \(r\) of the curve. (a) Write an equation that expresses this variation. (b) A car weighing 1600 lb travels around a curve at 60 \(\mathrm{mi} / \mathrm{h}\) . The next car to round this curve weighs 2500 \(\mathrm{lb}\) and requires the same force as the first car to keep from skidding. How fast is the second car traveling?

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Find an equation of the line that satisfies the given conditions. Through \((-2,-11) ;\) perpendicular to the line passing through \((1,1)\) and \((5,-1)\)

(a) Sketch the line with slope \(\frac{3}{2}\) that passes through the point \((-2,1)\) . (b) Find an equation for this line.

Find an equation of the line that satisfies the given conditions. y-intercept 6 : parallel to the line \(2 x+3 y+4=0\)
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