10.5 解决具有一面变量的激进等式

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  解决具有一个侧面变量的激进方程式

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  A common formula for a human's body surface area is  $\begin{array}{r}\text{BSA}=\sqrt{\frac{W\cdot H}{3600}},\end{array}$  where  W is weight in kilograms and  H is height in centimeters. If the average BSA for a man is 1.9 m ²  and the average height of a man is 175 centimeters, what is the average weight of a man in kilograms ¹ ? 
  ::人体表面面积的通用公式是BSA=WH3600,其中W是公斤重量,H是厘米高度。如果男性的平均BSA为1.9平方米,而男性的平均高度为175厘米,那么男性的平均重量是多少?

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  We need to solve an equation to answer this question—an equation with a square root . We see how to do that below. 
  ::我们需要解决一个解答这个问题的方程式——一个用平方根的方程式。

  

   

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  Solving Radical Equations
  ::解决激进等号

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    Steps for Solving a Radical Equation
  ::解决激进等同的步骤

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  1. Isolate the radical expression on one side of the equation.
  ::1. 孤立方程式一边的激进表达。

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  2. Use the power rule of exponents to undo the radical, that is, 
  ::2. 利用推手的权力规则来消除激进,即:

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  $$\begin{array}{rl}{a}^{\frac{1}{n}}& =b\\ (a^{\frac{1}{n}}{)}^n& =b^n\\ a& =b^n.\end{array}$$

  
  ::an=b(a1n)n=bna=bn。

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  3. Solve the equation.
  ::3. 解决等式。

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  4. Check for extraneous solutions.
  ::4. 寻找不相干的解决办法。

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  Example 1
  ::例1

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  Solve $\begin{array}{r}\sqrt{2x-5}+7=16\end{array}$ .
  ::解决 2x-5+7=16。

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  Solution: To solve for $\begin{array}{r}x\end{array}$ , we need to isolate the radical. Subtract 7 from both sides.
  ::解答: 要解决x, 我们需要分离激进分子。 从两边减去 7 。

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  $$\begin{array}{rl}\sqrt{2x-5}+7& =16\\ \sqrt{2x-5}& =9\end{array}$$

  
  ::2x-5+7=162x-5=9

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  Now, we can square both sides to undo the radical. 
  ::现在,我们可以让双方平分 来推翻激进派

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  $$\begin{array}{rl}{\sqrt{(2x-5)}}^2& =9^2\\ 2x-5& =81\\ 2x& =86\\ x& =43\end{array}$$

  
  :2x-5)2=922x-5=812x=86x=43

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  Check: $\begin{array}{r}\sqrt{2\left(43\right)-5}+7=\sqrt{86-5}+7=\sqrt{81}+7=9+7=16\end{array}$
  ::检查:2(43)-5+7=86-5+7=81+7=7=9+7=16

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     WARNING
  ::警告

  ALWAYS check your answers when solving radical equations. Sometimes you will solve one equation, get a solution, and then plug it back in, and it will not satisfy the equation. These types of "solutions" are called extraneous solutions and are not actually solutions. 

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  Example 2
  ::例2

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  Solve $\begin{array}{r}3\sqrt[3]{x-8}-2=\text{-}14\end{array}$ .
  ::解决 3x-83-2=14。

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  Solution: Again, isolate the radical 1st. Add 2 to both sides and divide by 3.
  ::解决方案:再次将激进的一号隔离开来。 双方加2号,分3号,分3号。

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  $$\begin{array}{rl}3\sqrt[3]{x-8}-2& =\text{-}14\\ 3\sqrt[3]{x-8}& =\text{-}12\\ \sqrt[3]{x-8}& =\text{-}4\end{array}$$

  
  ::3-83-2=-143x-83=-12-83=-4

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  Now, cube both sides to eliminate the radical.
  ::现在 立方两边消灭激进分子

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  $$\begin{array}{rl}{\sqrt[3]{(x-8)}}^3& =(\text{-}4{)}^3\\ x-8& =\text{-}64\\ x& =\text{-}56\end{array}$$

  
  :x-8)33=(-4)3x-8=-64x=-56)

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  Check: $\begin{array}{r}3\sqrt[3]{\text{-}56-8}-2=3\sqrt[3]{\text{-}64}-2=3\cdot \text{-}4-2=\text{-}12-2=\text{-}14\end{array}$
  ::检查:3-56-83-2=3-643-2=3--4-2=12-2=14

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  by Greg Stiffler demonstrates how to solve a radical equation with a cube root . 
  ::Greg Stiffler用立方根演示如何解析激进方程式。

   

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  Example 3
  ::例3

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  Solve  $\begin{array}{r}\sqrt[4]{3x+11}-2=3\end{array}$ .
  ::解决 3x+114-2=3。

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  Solution: In this problem, we have a 4th root. That means, once we isolate the radical, we must raise both sides to the 4th power to eliminate it.
  ::解决问题:在这个问题上,我们有一个第四根根。这意味着,一旦我们孤立了激进分子,我们必须将双方推向第四势力,以消灭它。

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  $$\begin{array}{rl}\sqrt[4]{3x+11}-2& =3\\ (\sqrt[4]{3x-11}{)}^4& =5^4\\ 3x-11& =625\\ 3x& =636\\ x& =212\end{array}$$

  
  ::3x+114-2=3(3x-114)4=543x-11=6253x=636x=212

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  Check: $\begin{array}{r}\sqrt[4]{3\left(212\right)+11}-2=\sqrt[4]{636-11}-2=\sqrt[4]{625}-2=5-2=3\end{array}$
  ::检查:3(212)+114-2=636-114-2=6254-2=5-2=3

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  Example 4
  ::例4

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  A common formula for a human's body surface area is $\begin{array}{r}BSA=\sqrt{\frac{W\cdot H}{3,600}},\end{array}$  where W is weight in kilograms and H is height in centimeters. If the average BSA for a man is 1.9 m ² and the average height of a man is 175 centimeters, what is the average weight of a man in kilograms ¹ ?   
  ::人体表面面积的通用公式是BSA=WH3,600,其中W是公斤重量,H是厘米高度。如果男性的平均BSA为1.9平方米,而男性的平均高度为175厘米,那么男性的平均重量是多少?

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  Solution:  Let's substitute the known values and then solve for W.
  ::解决方案:让我们取代已知的价值观,然后解决W。

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  $$\begin{array}{rl}1.9& =\frac{\sqrt{W\cdot 175}}{\sqrt{3,600}}\\ 1.9& =\frac{\sqrt{W}\cdot \sqrt{175}}{60}\\ 1.9& =.220\sqrt{W}\\ 8.636& =\sqrt{W}\\ (8.636{)}^2& =(\sqrt{W}{)}^2\\ 74.587& =W\end{array}$$

  The average man weighs about 74 kilograms.
  ::1.9=W 1753 6001.9=W 175601.9=W 220W8.636=W(8.636)2=(W)274.587=W 平均体重约74公斤。

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  by demonstrates an application of solving radical equations. 
  ::来证明解决激进方程式的应用。

   

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  Example 5
  ::例5

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  Solve  $\begin{array}{r}5\sqrt{2x-1}+1=26\end{array}$ .
  ::解决52x-1+1=26。

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  Solution: Isolate the radical by subtracting 1 and then dividing by 5.
  ::解决方案:将激进分子隔离,减去1,然后除以5。

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  $$\begin{array}{rl}5\sqrt{2x-1}-1& =\text{-}26\\ 5\sqrt{2x-1}& =\text{-}25\\ \sqrt{2x-1}& =\text{-}5\end{array}$$

  
  ::52-1-1=2652x-1=-252x-1=-252x-1=-5

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  Square both sides and continue to solve for $\begin{array}{r}x\end{array}$ .
  ::两边的广场 继续解决 x 。

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  $$\begin{array}{rl}(\sqrt{2x-1}{)}^2& =(\text{-}5{)}^2\\ 2x-1& =25\\ 2x& =26\\ x& =13\end{array}$$

  
  :2x-1)2=(5)22x-1=252x=26x=13)

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  Check: $\begin{array}{r}5\sqrt{2\left(13\right)-1}-1=5\sqrt{26-1}-1=5\sqrt{25}-1=5\cdot 5-1=25-1=24\end{array}$
  ::检查: 52(13) - 1 - 1=526 - 1=1=1=1=525 - 1=5_5=1=1=1=1=5=1=1=25=1=24

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  The solution does not check. Therefore , it is an extraneous solution . There are no solutions to this equation.
  ::解决方案无法校正。 因此, 它是一个不相干的解决办法 。 此方程式没有解决方案 。

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  Summary
  ::摘要

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  • To solve a radical equation, isolate the radical expression, use the power rule to undo the radical, solve the equation, and check for extraneous solutions.
    ::为了解决激进方程式, 孤立激进表达, 使用权力规则推翻激进, 解决方程式, 并检查不相干的解决办法。

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  Review
  ::回顾

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  Solve the equations and check for extraneous solutions.
  ::解决方程式,检查不相干的解决办法。

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  1. $\begin{array}{r}\sqrt{x+5}=6\end{array}$
  ::1. x+5=6

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  2. $\begin{array}{r}2-\sqrt{x+1}=0\end{array}$
  ::2 - x+1=0

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  3. $\begin{array}{r}4\sqrt{5-x}=12\end{array}$
  ::3-x=12

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  4. $\begin{array}{r}\sqrt{x+9}+7=11\end{array}$
  ::4. x+9+7=11

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  5. $\begin{array}{r}\frac{1}{2}\sqrt[3]{x-2}=1\end{array}$
  ::5. 12x-23=1

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  6. $\begin{array}{r}\sqrt[3]{x+3}+5=9\end{array}$
  ::6. x+33+5=9

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  7. $\begin{array}{r}5\sqrt{15-x}+2=17\end{array}$
  ::7. 515-x+2=17

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  8. $\begin{array}{r}\text{-}5=\sqrt[5]{x-5}-7\end{array}$
  ::8.5=x-55-7

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  9. $\begin{array}{r}\sqrt[4]{x-6}+10=13\end{array}$
  ::9. x-64+10=13

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  10. $\begin{array}{r}\frac{8}{5}\sqrt[3]{x+5}=8\end{array}$
  ::10. 85x+53=8

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  11. $\begin{array}{r}3\sqrt{x+7}-2=25\end{array}$
  ::11. 3x+7-2=25

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  12. $\begin{array}{r}\sqrt[4]{235+x}+9=14\end{array}$
  ::12. 235+x4+9=14

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  Explore More
  ::探索更多

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  1.  The legs of a right triangle measure 3 and $\begin{array}{r}2\sqrt{x}\end{array}$ . The hypotenuse measures 5. What is the length of the leg with the unknown value?
  ::1. 右三角形的腿3和2x. 下限测量5. 腿长多少值未知?

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  2.  If the surface area of a sphere is the same as the curved surface area of a right circular cylinder whose height and diameter are 12 cm each, then what is the radius of the sphere?
  ::2. 如果一个球体的表面面积与右圆圆柱形的弯曲表面积相同,其高度和直径各为12厘米,那么球体的半径是多少?

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  3. If the height of a triangle is twice its base and area of the triangle is 36 cm ² , then what are its base and height?  
  ::3. 如果三角形的高度是其基数的两倍,三角形的面积是36厘米2,那么其基数和高度是多少?

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  4. A rectangular picture is 9 inches wide and 12 inches long. The picture has a frame of uniform width. If the combined area of picture and frame is 180 in ² , what is the width of the frame?
  ::4. 矩形图象宽9英寸,长12英寸,有统一宽度框,如果图像和框架的合并区域是180英寸2,则框架宽度是多少?

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  5. A computer screen has a diagonal of 28.8 inches and a length of 14.3 inches. Determine its width.
  ::5. 计算机屏幕的对角为28.8英寸,长度为14.3英寸,确定宽度。

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  6. The motion of an object is modeled by the function $\begin{array}{r}s=2.3t^4,\end{array}$  where $\begin{array}{r}t\end{array}$  is the time (measured in seconds) and $\begin{array}{r}s\end{array}$  is the position of this object (measured in meters). Find the time when $\begin{array}{r}s=\end{array}$  12.4 m. Round your answer to three decimal points.
  ::6. 对象的动作由函数 s=2.3t4 模拟, t 是时间( 以秒计) , s 是该对象的位置( 以米计) 。 查找秒= 12. 4米的时间 。 您的回答四舍五入到小数点后三点 。

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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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  PLIX
  ::PLIX

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  Try these interactives that reinforce the concepts explored in this section:
  ::尝试这些强化本节所探讨概念的交互作用 :

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  References
  ::参考参考资料

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  1. "Body Surface Area," last edited May 22, 2017,
  ::1. 2017年5月22日 上次编辑的“生物表面区域”