8.3 利用平根解决赤道赤道

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  利用平根解决二次赤道

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  Suppose Tom wants to prank his friend by hitting him with a water balloon on a hot summer day. If the balloon is dropped from the 6th floor of an apartment building that is 70 feet high, after how many seconds will the balloon hit the friend?
  ::假设汤姆想在炎热的夏日用水气球打他的朋友来恶作剧。 如果气球是从一个高达70英尺的公寓大楼的6楼掉下来的,那么气球会用几秒钟打到他的朋友呢?

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  We can determine this using square roots, and we'll discuss how in this section. 
  ::我们可以用平方根来决定这一点, 我们将讨论如何在这个部分。

  

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  Solving Quadratic Equ ations by Using  Square Roots 
  ::使用平根解决赤道等量

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  Another technique for solving quadratic equations is the square root method. If we can isolate a squared term on one side of the equation , we can apply this method. 
  ::解决二次方程式的另一种技术是平方根法。如果我们能在方程式的一边隔离一个正方字,我们就可以应用这种方法。

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     Square Root Method
  ::平方根法

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  If   $\begin{array}{r}{a}^2=b\end{array}$ , then   $\begin{array}{r}a=\pm \sqrt{b}\end{array}$ .  
  ::如果 a2=b, 那么 a*b.

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  WARNINGS:
  ::警告:

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  • Remember to include the plus or minus sign to indicate the  positive and negative roots. 
    ::记住要包括增减符号,以表明正反两方面的根源。
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  • To use this method there can be only  an  $\begin{array}{r}{x}^2\end{array}$  and a constant term; there can't be an  x- term. 
    
    ::要使用此方法, 只能使用 x2 和 常数条件; 无法使用 x 期 。

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  Example 1
  ::例1

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  Solve $\begin{array}{r}2x^2-3=15\end{array}$ .
  ::解决 2x2 -3=15。

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  Solution:  To use the square root method, we need to start by isolating the $\begin{array}{r}{x}^2\end{array}$ .
  ::解决方案: 要使用平方根法, 我们首先需要分离 x2 。

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  $$\begin{array}{rl}2x^2-3& =15\\ 2x^2& =18\\ x^2& =9\end{array}$$

  
  ::2x2-3=152x2=18x2=9

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  At this point, you can take the square root of both sides.
  ::在这一点上,你可以选择双方的平方根。

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  $$\begin{array}{rl}\sqrt{x^2}& =\pm \sqrt{9}\\ x& =\pm 3\end{array}$$

  
  ::x29x3

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  Notice that $\begin{array}{r}x\end{array}$ has two solutions; 3 or -3. When taking the square root of both sides, always put the $\begin{array}{r}\pm \end{array}$ (plus or minus sign) in front of the square root. 
  ::注意 x 有两个解决方案; 3 或 - 3. 选择两侧的平方根时, 总是将 +( + 或 减 符号) 放在平方根前面 。

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  Checking the solutions yields
  ::检查解决方案效果

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  $$\begin{array}{rl}& 2(3{)}^2-3=152(\text{-}3{)}^2-3=15\\ & 2\cdot 9-3=15\text{or}2\cdot 9-3=15\\ & 18-3=1518-3=15\end{array}$$

  
  ::2(3)2-3=152(-3)2-3=15 2__9-3=15or 2__9-3=15or 2__9-3=3=1518-3=1518-3=1518-3=15

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  This video by CK-12 demonstrates how to by using the square root method.
  ::CK-12的这段影片展示了如何使用平方根法。

   

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  Example 2
  ::例2

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  Tom  wants to prank his friend by hitting him with a water balloon on a hot summer day. If the balloon is dropped from the 6th floor of an apartment building that is 70 feet high, after how many seconds will the balloon hit the friend? Assume the friend is 6 feet tall.
  ::汤姆想在炎热的夏日用水球打他的朋友,以此来欺骗他的朋友。如果气球是从一座70英尺高的公寓大楼的6楼掉下来的,那么气球会打到他的朋友几秒钟?假设他的朋友是6英尺高。

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  Solution: The water balloon will be in free fall for 64 feet, which is the height difference between the apartment window and the friend. This gives us an equation of  $\begin{array}{r}h=\text{-}32t^2+70\end{array}$ .  We want to consider when  $\begin{array}{r}h=6\end{array}$ .
  ::溶液: 水气球将自由下落64英尺, 这是公寓窗户和朋友之间的高度差。 这让我们可以得出 h=- 32t2+70 的方程式。 我们想在h=6时考虑。

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  $$\begin{array}{r}6=\text{-}32t^2+70\\ 0=\text{-}32t^2+64\\ \text{-}64=\text{-}32t^2\\ 2=t^2\\ \pm \sqrt{2}=t\end{array}$$

  
  ::6=-32t2+700=-32t2+64-64=-32t22=-t2%2=t

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  We reject the negative solution because in this context it makes no sense. The time it takes to hit the friend is  $\begin{array}{r}\sqrt{2}\approx 1.4141\end{array}$  seconds.   
  ::我们拒绝否定的解决办法,因为在这种情况下,它没有意义。 击打朋友的时间是21.4141秒。

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  Example 3
  ::例3

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  Solve  $\begin{array}{r}11+x^2=4x^2+5\end{array}$ .
  ::解决 11+x2=4x2+5。

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  Solution: Combine all like terms , then isolate $\begin{array}{r}{x}^2.\end{array}$
  ::解答: 将所有类似术语合并, 然后分离 x2 。

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  $$\begin{array}{rl}11+x^2& =4x^2+5\\ \text{-}3x^2& =\text{-}6\\ x^2& =2\\ x& =\pm \sqrt{2}\end{array}$$

  
  ::11+x2=4x2+5-3x2=6x2=2x%2

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  Example 4
  ::例4

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  Solve $\begin{array}{r}\frac{x^2}{16}+3=27\end{array}$ .
  ::解决 x216+3=27 。

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  Solution: Isolate $\begin{array}{r}{x}^2\end{array}$ and then take the square root.
  ::溶液: 隔离 x2, 然后取平方根 。

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  $$\begin{array}{rl}\frac{x^2}{16}+3& =27\\ \frac{x^2}{16}& =24\\ x^2& =384\\ x& =\pm \sqrt{384}=\pm 8\sqrt{6}\end{array}$$

  
  ::x216+3=27x216=24x2=384x38486

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  Example 5
  ::例5

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  Solve $\begin{array}{r}3(x-5{)}^2+7=43\end{array}$ .
  ::解决(3x-5)2+7=43。

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  Solution: In this example, $\begin{array}{r}x\end{array}$ is not the only thing that is squared. Isolate the $\begin{array}{r}(x-5{)}^2\end{array}$ and then take the square root.
  ::解决方案 : 在此示例中, x 不是唯一的正方形 。 分离 (x-5) 2, 然后取平方根 。

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  $$\begin{array}{rl}3(x-5{)}^2+7& =43\\ 3(x-5{)}^2& =36\\ (x-5{)}^2& =12\\ x-5& =\pm \sqrt{12}\text{ or}\pm 2\sqrt{3}\end{array}$$

  
  ::3(x--5)2+7=433(x-5)2=36(x-5)2=12x-512或23

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  Add 5 to both sides.
  ::双方加5个。

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  $$\begin{array}{rl}x-5& =\pm 2\sqrt{3}\\ x& =5\pm 2\sqrt{3}\end{array}$$

  
  ::x-523x=523

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  $\begin{array}{r}x=5+2\sqrt{3}\end{array}$ or $\begin{array}{r}5-2\sqrt{3}\end{array}$ . We can estimate these solutions as decimals: 8.46 or 1.54. 
  ::x=5+23或5-23。我们可以以小数点:8.46或1.54来估计这些溶液。

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  by CK-12 demonstrates an application of how to solve quadratic equations by using the square root method.
  ::通过 CK-12 演示应用如何使用平方根法解析二次方程。

   

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  Feature: Free Falling
  ::特色:自由坠落

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  by Deirdre Mundy
  ::由Deirdre Mundy 编辑

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  When does a normal jet plane help researchers study space? When it flies in parabolas! Space programs throughout the world use specially trained pilots to test the effects of weightlessness on people, plants, animals, and equipment.
  ::普通喷气式飞机何时能帮助研究人员研究空间?当它飞到抛物板时!世界各地的空间方案都使用经过专门训练的飞行员来测试失重对人、植物、动物和设备的影响。

   

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  A Quadratic Equation for Weightlessness
  ::无轻度的赤道等同

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  Recall that the graph of a quadratic equation is parabolic. During a parabolic flight, people and objects on a plane alternately experience high gravity and low gravity. The amount of gravity they experience depends on the pilot's flight path: the steeper the parabola , the higher the highs and the lower the lows. In a typical parabolic maneuver, the plane accelerates rapidly while traveling upward at a 45- degree angle. At the top of its climb, the pilot allows the plane to enter into free fall, during which every person and thing on board the aircraft experiences the feeling of weightlessness, floating away from the floor. The plane's motion creates an illusion of "zero gravity." The period of zero gravity only lasts about 20 to 30 seconds at a time, so the planes usually fly 30 or more parabolic maneuvers in a single flight. Because of the high potential for motion sickness in the air, these planes have been dubbed "vomit comets."
  ::回想一下方形的图形是抛物线。 在抛物线飞行期间, 飞机上的人和物体交替经历高重力和低重力。 他们经历的重力程度取决于飞行员的飞行路径: 抛物线越陡峭, 高度越高, 低层越低。 在典型的抛物线操作中, 飞机在以45度角度向上移动时加速加速。 在飞行的顶端, 飞行员允许飞机自由坠落, 在此期间, 飞机上每个人和物体都感到无重力, 漂浮在地上。 飞机的动作产生了“ 零重力” 的幻觉。 零重力时长大约20至30秒, 因此, 飞机通常在单次飞行中飞行30次或30次以上。 由于飞机在空中运动生病的可能性很大, 这些飞机被调出“ 喷射彗星 ” 。

  

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  You do not have to be an astronaut or a scientist to experience parabolic flight. NASA sponsors a Reduced Gravity Education Flight Program, open to students and teachers who design and propose experiments to conduct on board. It is also possible to buy tickets for commercial flights on reduced gravity aircraft. The European Space Agency sells tickets for rides on its parabolic flights. The same zero gravity planes that NASA uses sometimes tour the United States. Tourists in cities like Las Vegas can spend $5,000 a piece for a chance to experience weightlessness aboard a parabolic flight.
  ::你不必是宇航员或科学家才能经历抛物线飞行。美国航天局赞助了一个降低重力教育飞行方案,向设计并提议在飞机上进行实验的学生和教师开放。也可以购买使用减少重力飞机的商业飞行的机票。欧洲航天局出售其抛物线飞行的机票。美国航天局有时使用同样的零重力飞机巡航美国。拉斯维加斯等城市的旅游者可以花5 000美元,一次在抛物线飞行上体验失重的机会。

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  by the BBC  explains zero gravity flight. 
  ::英国广播公司解释零重力飞行。

   

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  Summary
  ::摘要

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  • To solve quadratic equations by the square root method, isolate the squared term and the constant term on opposite sides of the equation. Then take the square root of both sides, making the side with the constant term plus or minus the square root.  
    ::以平方根法解析二次方程, 将正方词和常数分隔在方程的对面。 然后取两边的平方根, 用常数加上或减去平方根使侧面成为常数加上或减去平方根 。

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  Review
  ::回顾

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  Solve the following quadratic equations. Simplify answers as much as possible.
  ::解决以下的二次方程。 尽可能简化答案 。

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  1.  $\begin{array}{r}{x}^2=144\end{array}$
  ::1. x2=144

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  2.  $\begin{array}{r}5x^2-4=16\end{array}$
  ::2. 5x2-4=16

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  3.  $\begin{array}{r}8-10x^2=\text{-}22\end{array}$
  ::3. 8-10x2=-22

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  4.  $\begin{array}{r}(x+2{)}^2=49\end{array}$
  ::4. (x+2)2=49

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  5.  $\begin{array}{r}6(x-5{)}^2+1=19\end{array}$
  ::5. 6(xx-5-5)2+1=19

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  6.  $\begin{array}{r}\frac{3}{4}{x}^2-19=26\end{array}$
  ::6. 34x2-19=26

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  7.  $\begin{array}{r}{x}^2-12=36-2x^2\end{array}$
  ::7. x2 - 12=36-2x2

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  8.  $\begin{array}{r}9-\frac{x^2}{3}=\text{-}33\end{array}$
  ::8. 9-x23=-33

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  9.  $\begin{array}{r}\text{-}4(x+7{)}^2=\text{-}52\end{array}$
  ::9.4(x+7)2=-52

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  10.  $\begin{array}{r}2(3x+4{)}^2-5=45\end{array}$
  ::10. 2(3x+4)2-5=45

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  11.  $\begin{array}{r}\frac{1}{3}(x-10{)}^2-8=16\end{array}$
  ::11. 13(x-10)2-8=16

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  12.  $\begin{array}{r}\frac{(x-1{)}^2}{6}-\frac{8}{3}=\frac{7}{2}\end{array}$
  ::12. (x-1)26-83=72

   

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  Explore More 
  ::探索更多

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  1. The aspect ratio of a TV screen is the ratio of the screen's width to its height. For HDTVs, the aspect ratio is 16:9 ¹ . What is the width and height of a 42-inch-screen TV? (The 42 inches refers to the length of the screen's diagonal.) Since the TV is a rectangle, the length, width, and diagonal form a right triangle. Therefore, you can use the Pythagorean Theorem. Round your answers to the nearest hundredth.
  ::1. 电视屏幕的宽度比是屏幕宽度与高度之比。对于HDTV来说,宽度比是16:91。42英寸屏幕电视的宽度和高度是多少?(42英寸是屏幕对角的长度。 )由于电视是一个矩形,长度、宽度和对角形形成一个右三角形。因此,您可以使用 Pythagoren Theorem 。您可以将答案绕到最近的一百个答案 。

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  2. Victor drops an apple out of a  10th-floor  window, which is 120 feet above the ground. One second later, Juan drops an orange out of a 6th-floor window, which is 72 feet above the ground. Which fruit reaches the ground 1st? What is the time difference between the fruits' arrival on the ground?
  ::2. Victor从10楼的窗户上掉一个苹果,该窗户在地面上120英尺处;Juan从6楼的窗户上掉一个橙子,该窗户在地面上72英尺处;什么水果到达地面1号?水果到达地面之间的时间差是多少?

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  This video by CK-12 demonstrates how to solve application problems involving quadratic equations by using the square root method. 
  ::CK-12的这段影片展示了如何使用平方根方法解决涉及二次方程式的应用问题。

   

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  3. The area of a square plot of land is 3,600 square feet. What is the length of one side of the plot of land? 
  ::3. 平方土地面积为3,600平方英尺,土地面积一面的长度是多少?

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  4. The area of the top of a cylindrical canister is  $\begin{array}{r}49\pi \end{array}$  square inches. What is the diameter?
  ::4. 圆柱体罐顶部面积为49-平方英寸,直径是多少?

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  This video by CK-12 demonstrates how to solve application problems involving quadratic equations by using the square root method.
  ::CK-12的这段影片展示了如何使用平方根方法解决涉及二次方程式的应用问题。

   

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  5. a. Solve $\begin{array}{r}{x}^2-9=0\end{array}$  by factoring and by using square roots. Which do you think is easier? Why?
  ::5. a. 通过乘数和用平方根解决 x2-9=0。你认为哪一种比较容易?为什么?

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  b. Solve $\begin{array}{r}(3x-2{)}^2+1=17\end{array}$  by using square roots. Then, solve $\begin{array}{r}3x^2-4x-4=0\end{array}$   by factoring. What do you notice? What can you conclude?
  ::b. 使用平方根解决(3x-2-2)2+1=17。然后通过乘数解决 3x2-4x-4=0。你注意到什么?你能得出什么结论?

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  Answers for Review and Explore More Problems
  ::回顾和探讨更多问题的答复

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  Please see the Appendix. 
  ::请参看附录。

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  References
  ::参考参考资料

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  1. "Understanding Aspect Ratio - Crutchfield," by Steve Kindig, last accessed May 26, 2017,
  ::1. “了解视线比率-Crutchfield”,由Steve Kindig撰写,2017年5月26日最后一次访问,