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Chapter 8: Problem 64

Solve by completing the square. See Section 8.1.$$ x^{2}+14 x+20=0 $$

Short Answer

Expert verified
The solutions are \(x = -7 + \sqrt{29}\) and \(x = -7 - \sqrt{29}\).

Step by step solution

01

Move the Constant Term

First, we need to move the constant term to the other side of the equation to begin the process of completing the square. Starting with: \[ x^2 + 14x + 20 = 0 \]Subtract 20 from both sides:\[ x^2 + 14x = -20 \]
02

Find the Number to Complete the Square

We will complete the square for the expression on the left, \(x^2 + 14x\). Take half of the coefficient of \(x\) (which is 14), square it, and add it to both sides of the equation. Half of 14 is 7, and squaring it gives 49:\[ \left( \frac{14}{2} \right)^2 = 49 \]Add 49 to both sides:\[ x^2 + 14x + 49 = -20 + 49 \]
03

Write as a Square

Now, the left side of the equation is a perfect square trinomial. It can be factored as:\[ (x + 7)^2 = 29 \]
04

Solve for x

Take the square root of both sides to solve for \(x\):\[ x + 7 = \pm \sqrt{29} \]Next, solve for \(x\) by subtracting 7 from both sides:\[ x = -7 \pm \sqrt{29} \]

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Key Concepts

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

Solving Quadratic Equations
Quadratic equations are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. Solving these equations is a crucial skill in algebra. There are various ways to solve them, such as factoring, using the quadratic formula, or completing the square. Completing the square is a technique used to transform a quadratic expression into a perfect square trinomial. This method is particularly useful when factoring is not straightforward or when the coefficient of \(x^2\) is 1.
  • Move the constant term to the opposite side of the equation. This simplifies the expression so that you can focus on the quadratic and linear terms.
  • Find a number to add to both sides of the equation to create a perfect square trinomial.
  • Rewrite the trinomial as the square of a binomial, making it easier to solve for \(x\).
Applying the method of completing the square often provides a clear path to finding the roots of the equation.
Perfect Square Trinomial
A perfect square trinomial is an expression that results in the square of a binomial. It's important in the process of completing the square, as it allows us to rewrite a quadratic expression in a more manageable form. The general form of a perfect square trinomial is \((x + p)^2 = x^2 + 2px + p^2\).
To transform a quadratic expression into a perfect square trinomial:
  • Focus on the terms with \(x^2\) and \(x\). Ignore the constant term initially.
  • Take half of the coefficient of \(x\), which in this specific exercise example is 14, giving 7.
  • Square the result, here resulting in 49, and add it to both sides of your equation.
The left side of the equation can then be factored into \((x + 7)^2\). This process helps simplify further steps needed to solve the quadratic equation.
Square Root Method
The square root method provides an efficient way to solve equations once they are expressed as squares, like after completing the square. When you have an equation like \((x + 7)^2 = 29\), you can proceed by taking the square root of both sides.
To use this method effectively:
  • Take the square root of both sides of the equation. Remember to consider both the positive and negative roots, because squaring a number makes it positive.
  • This will yield two solutions: \(x + 7 = \sqrt{29}\) and \(x + 7 = -\sqrt{29}\).
  • Subtract 7 from both sides in each equation to solve for \(x\).
Thus, you find \(x = -7 + \sqrt{29}\) and \(x = -7 - \sqrt{29}\). Using the square root method after completing the square provides a clear answer and illustrates the solutions visually on a graph.

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Most popular questions from this chapter

Find the maximum or minimum value of each function. Approxi- mate to two decimal places. The projected number of Wi-Fi-enabled cell phones in the United States can be modeled by the quadratic function \(c(x)=-0.4 x^{2}+21 x+35,\) where \(c(x)\) is the projected number of Wi-Fi-enabled cell phones in millions and \(x\) is the number of years after \(2009 .\) (Source: Techcrunchies.com) A. Will this function have a maximum or a minimum? How can you tell? B. According to this model, in what year will the number of Wi-Fi-enabled cell phones in the United States be at its maximum or minimum? C. What is the maximum/minimum number of Wi-Fienabled cell phones predicted? Round to the nearest whole million.

Find the vertex of the graph of each quadratic function. Determine whether the graph opens upward or downward, find any intercepts, and sketch the graph. See Examples 1 through 4 . $$ f(x)=-5 x^{2}+5 x $$

Sketch the graph of each function. See Section 8.5. $$ f(x)=3(x-4)^{2}+1 $$

A common equation used in business is a demand equation. It expresses the relationship between the unit price of some commodity and the quantity demanded. For Exercises III and 112 , p represents the unit price and x represents the quantity demanded in thousands. Acme, Inc., sells desk lamps and has found that the demand equation for a certain style of desk lamp is given by the equation \(p=-x^{2}+15 .\) Find the demand for the desk lamp if the price is \(\$ 7\) per lamp.

Solve by completing the square. See Section 8.1. $$ z^{2}-8 z=2 $$
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