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Chapter 3: Problem 138

Will help you prepare for the material covered in the next section. Solve: \(x(x-7)=3\)

Short Answer

Expert verified
The solutions to the equation \(x(x-7)=3\) are \(x = (7 + \sqrt{61}) / 2\) and \(x = (7 - \sqrt{61}) / 2\)

Step by step solution

01

Rearrange the equation into standard form

The equation given is \(x(x-7)=3\). Distribute the x into the parenthesis to get \(x^2 - 7x = 3\). Subtract 3 from both sides to forms a quadratic equation \(x^2 - 7x - 3 = 0\). This express the given equation into the standard form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a=1\), \(b=-7\) and \(c=-3\).
02

Apply the quadratic formula

The quadratic formula is given by \(-b \pm \sqrt{b^2 - 4ac} / 2a\). Substituting \(a=1\), \(b=-7\) and \(c=-3\) into the quadratic formula will give us the roots of the equation. Thus, the values of x will be \(x = [-(-7) \pm \sqrt{(-7)^2 - 4*1*-3}] / 2*1\).
03

Solve for x

Calculate inside the square root first: \(49 + 12 = 61\). Then, compute the numerator: \(7 \pm \sqrt{61}\), and the denominator which is 2. Divide the results of the numerator by the denominator. The roots or solutions of the quadratic equation are \(x = (7 + \sqrt{61}) / 2\) and \(x = (7 - \sqrt{61}) / 2\).

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Most popular questions from this chapter

Will help you prepare for the material covered in the next section. $$\text { Solve: }(x-3)^{2} > 0$$

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

What is an exponential function?

Rewrite the equation in terms of base \(e\). Express the answer in terms of a natural logarithm and then round to three decimal places. $$y=2.5(0.7)^{x}$$

Suppose that a population that is growing exponentially increases from 800,000 people in 2010 to 1,000,000 people in \(2013 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.
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