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Chapter 1: Problem 57

Solve each equation using a graphing calculator. [Hint: Begin with the window \([-10,10]\) by \([-10,10]\) or another of your choice (see Useful Hint in Graphing Calculator Terminology following the Preface) and use ZERO, SOLVE, or TRACE and ZOOM IN.] (In Exercises 61 and 62 , round answers to two decimal places.) $$ 4 x^{2}+24 x+45=9 $$

Short Answer

Expert verified
The solutions are both approximately x = -3.

Step by step solution

01

Rearrange the equation

First, we need to rewrite the equation by moving everything to one side so it equals zero. We have: \[4x^2 + 24x + 45 = 9\]Subtract 9 from both sides:\[4x^2 + 24x + 45 - 9 = 0\]This simplifies to:\[4x^2 + 24x + 36 = 0\]
02

Identify the function for graphing

We will use a graphing calculator to solve the equation. Consider the function:\[f(x) = 4x^2 + 24x + 36\]We need to find the x-values where this function crosses the x-axis, which represents the roots of the equation.
03

Set up the graphing window

Open your graphing calculator and set the window to an appropriate size to easily find the roots. A good starting point is: Xmin = -10, Xmax = 10, Ymin = -10, Ymax = 10.
04

Graph the equation

Enter the function \(f(x) = 4x^2 + 24x + 36\) into your graphing calculator and plot the graph. Observe where the curve intersects the x-axis.
05

Find the x-intercepts

Use the "ZERO" function on your calculator to find the points where the graph crosses the x-axis. These points are the solutions to the equation.
06

Read and record the solutions

After finding the x-intercepts using the "ZERO" function, record these x-values. The solutions are where the equation \(4x^2 + 24x + 36 = 0\) equals zero.
07

Round the solutions

You need to round the x-values to two decimal places. Let's say the solutions are x = -3 and x = -3 (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.

Quadratic Equations
Quadratic equations are fundamental in mathematics and often appear in algebra. These are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\). The highest degree of the variable \(x\) is 2, hence the term "quadratic" referring to the square nature of the equation. These equations form a parabolic shape when graphed.

When solving quadratic equations, one can use various methods:
  • Factoring: This involves expressing the quadratic as a product of binomials, if possible.
  • Completing the Square: A technique to convert the equation into a perfect square trinomial.
  • Quadratic Formula: Utilizing the formula \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\), which directly finds the solutions.
  • Graphing: Using graphing tools or calculators to visually determine the solutions, which are the points where the parabola crosses the x-axis.
Each method offers its benefits depending on the structure of the quadratic equation, with graphing being particularly intuitive for visual learners.
X-Intercepts
X-intercepts are the points where a graph crosses the x-axis. In the context of a quadratic equation like \(4x^2 + 24x + 36 = 0\), the x-intercepts are the solutions to the equation. These intercepts represent the values of \(x\) that make the function equal to zero.

To find x-intercepts using a graphing calculator, it's important to:
  • Enter the quadratic function into the calculator.
  • Select a suitable viewing window to properly see where the graph crosses the x-axis.
  • Utilize features such as "ZERO" or "SOLVE" to pinpoint the intercepts precisely.
The x-intercepts are crucial because they reveal the real roots of the quadratic equation. In our specific example, after graphing and using the calculator's tools, we found that the x-intercepts for \(f(x) = 4x^2 + 24x + 36\) are both at \(x = -3\). These points indicate where the function's value is zero.
Graphing Functions
Graphing functions such as quadratics can reveal much about the behavior and nature of the equations they represent. A graphing calculator is an excellent tool for this task, as it offers a visual way to solve and understand equations.

For success in graphing a quadratic function:
  • First rewrite the equation to a standard form if needed; this means all terms on one side equaling zero.
  • Setup your calculator's display window appropriately. A common choice is \([-10, 10]\) for both your x and y ranges.
  • Enter the quadratic function into the calculator to create a plot.
  • Analyze the graph, focusing on points of interest like the vertex and x-intercepts.
Using the graphing technique allows students to visualize the entire curve. This not only helps identify intersections with the x-axis but also presents key features like the vertex and symmetry of the parabola.

In practical use, this visual representation can make problem-solving more intuitive for learners, as graphing offers immediate insight into the function's behavior.

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Most popular questions from this chapter

Find the slope (if it is defined) of the line determined by each pair of points. \(\left(-2, \frac{1}{2}\right)\) and \(\left(5, \frac{1}{2}\right)\)

The world population (in millions) since the year 1700 is approximated by the exponential function \(P(x)=522(1.0053)^{x}\), where \(x\) is the number of years since 1700 (for \(0 \leq x \leq 200\) ). Using a calculator, estimate the world population in the year: $$ 1800 $$

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h}\). (Assume \(h \neq 0 .\) ) $$ \begin{array}{l} f(x)=x^{4} \\ \text { [Hint: Use }(x+h)^{4}=x^{4}+4 x^{3} h+6 x^{2} h^{2}+ \\ \left.4 x h^{3}+h^{4} .\right] \end{array} $$

For each equation, find the slope \(m\) and \(y\) intercept \((0, b)\) (when they exist) and draw the graph. \(y=-\frac{1}{3} x+2\)

For each pair of functions \(f(x)\) and \(g(x)\), find a. \(f(g(x))\) b. \(g(f(x))\) and c. \(f(f(x))\) $$ f(x)=\sqrt{x} ; g(x)=x^{3}-1 $$
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