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Chapter 3: Problem 42

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-3)=2$$

Short Answer

Expert verified
Solutions: \( x = \frac{3 + \sqrt{17}}{2} \) and \( x = \frac{3 - \sqrt{17}}{2} \).

Step by step solution

01

Expand the Equation

Starting with the equation given, expand the left-hand side: \(x(x-3) = x^2 - 3x\). So, the equation becomes \(x^2 - 3x = 2\).
02

Move all terms to one side

Subtract 2 from both sides of the equation to set it to zero: \(x^2 - 3x - 2 = 0\). Now the equation is in a standard quadratic form.
03

Apply the Quadratic Formula

The standard form is \(ax^2 + bx + c = 0\), so \(a = 1\), \(b = -3\), \(c = -2\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
04

Calculate the Discriminant

Find the discriminant \(b^2 - 4ac\): \((-3)^2 - 4 \cdot 1 \cdot (-2) = 9 + 8 = 17\). Since the discriminant is positive, there are two real solutions.
05

Solve for x using the Quadratic Formula

Substitute the values into the quadratic formula: \[ x = \frac{-(-3) \pm \sqrt{17}}{2 \cdot 1} = \frac{3 \pm \sqrt{17}}{2} \]. So, the solutions are \( x = \frac{3 + \sqrt{17}}{2} \) and \( x = \frac{3 - \sqrt{17}}{2} \).
06

Graphical Representation

Graph the function \(f(x) = x^2 - 3x - 2\) and find the x-intercepts. These intercepts occur where \(f(x) = 0\), confirming the solutions \( x = \frac{3 + \sqrt{17}}{2} \) and \( x = \frac{3 - \sqrt{17}}{2} \).

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool used to solve quadratic equations. A quadratic equation is an equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
To find the roots (solutions) of the equation, you can use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's how it works step-by-step:
  • Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
  • Plug these values into the quadratic formula.
  • Solve for \(x\) to find the possible solutions.
Understanding this formula can greatly simplify the process of solving quadratic equations, especially when factoring is challenging.
Discriminant
The discriminant is a crucial part of the quadratic formula, found under the square root sign. It is given by
\(b^2 - 4ac\).
The discriminant helps in determining the nature of the solutions of a quadratic equation.
  • If the discriminant is positive (>0), there are two distinct real solutions.
  • If the discriminant is zero, there is exactly one real solution (the roots are the same).
  • If the discriminant is negative (<0), there are no real solutions (the solutions are complex numbers).
In our example, the discriminant is 17, which is positive, indicating two real solutions.
Real Solutions
Real solutions are the x-values where a quadratic equation equals zero. They can be visualized as the points where the graph of the quadratic function crosses the x-axis on a coordinate plane.
These points are also known as the roots or x-intercepts of the function.
  • To find the real solutions, compute them using the quadratic formula.
  • If the discriminant is positive, expect two different real numbers as solutions, like in the example, \(x = \frac{3 + \sqrt{17}}{2}\) and \(x = \frac{3 - \sqrt{17}}{2}\).
Real solutions are essential for understanding the behavior of quadratic functions across different intervals of x-values.
Graphical Representation
The graphical representation of a quadratic function helps visualize its solutions.
The graph of a quadratic equation forms a parabola. This parabola can either open upwards or downwards, depending on the sign of the coefficient \(a\).Here is what to look for when graphing:
  • Find the x-intercepts of the parabola, as these represent the real solutions.
  • The vertex of the parabola gives insight into the maximum or minimum value of the function.
  • Notice the axis of symmetry that divides the parabola into mirror images.
In our example, the quadratic function \(f(x) = x^2 - 3x - 2\) has x-intercepts at \(x = \frac{3 + \sqrt{17}}{2}\) and \(x = \frac{3 - \sqrt{17}}{2}\), confirming the solutions through the visual crossing of the x-axis.

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Most popular questions from this chapter

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(2 x^{2}-x+3<0\) (b) \(2 x^{2}-x+3 \geq 0\)

Find the complex conjugate. $$\frac{3}{-2 i}$$

Solve each problem. The coast-down time \(y\) for a typical car as it drops \(10 \mathrm{mph}\) from an initial speed \(x\) depends on several factors, such as average drag, tire pressure, and whether the transmission is in neutral. The table gives the coast-down time in seconds for a car under standard conditions for selected speeds in miles per hour. $$\begin{array}{|c|c|} \hline \begin{array}{c} \text { Initial Speed } \\ \text { (in mph) } \end{array} & \begin{array}{c} \text { Coast-Down } \\ \text { Time (in seconds) } \end{array} \\ \hline 30 & 30 \\ 35 & 27 \\ 40 & 23 \\ 45 & 21 \\ 50 & 18 \\ 55 & 16 \\ 60 & 15 \\ 65 & 13 \\ \hline \end{array}$$ (a) Plot the data. (b) Use the quadratic regression feature of a graphing calculator to find the quadratic function \(g\) that best fits the data. Graph this function in the same window as the data. Is \(g\) a good model for the data? (c) Use \(g\) to predict the coast-down time, to the nearest second, at an initial speed of 70 mph. (d) Use the graph to find the speed that comesponds to a coast-down time of 24 seconds.

Solve each formula for the indicated variable. Leave \(\pm\) in answers when applicable. Assume that no denominators are 0 $$s=\frac{1}{2} g t^{2} \quad \text { for } t$$

Solve each quadratic equation by completing the square. $$x^{2}-2 x=2$$
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