Section: 解决具有一个侧面变量的激进方程式 | 10.5 解决具有一面变量的激进等式

Section outline

A common formula for a human's body surface area is $\begin{aligned} BSA = \sqrt{\frac{W \cdot H}{3600}}, \end{aligned}$ 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?

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。

Solving Radical Equations
::解决激进等号

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

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

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:

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

::an=b(a1n)n=bna=bn。

3. Solve the equation.
::3. 解决等式。

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

Example 1
::例1

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

Solution: To solve for $\begin{aligned} x \end{aligned}$ , 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 。

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

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

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

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

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

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

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.

Example 2
::例2

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

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。

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

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

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

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

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

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

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。

Example 3
::例3

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

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。

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

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

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

Example 4
::例4

A common formula for a human's body surface area is $\begin{aligned} B S A = \sqrt{\frac{W \cdot H}{3, 600}}, \end{aligned}$ 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?

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。

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

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

Example 5
::例5

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

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。

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

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

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

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

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

Check: $\begin{aligned} 5 \sqrt{2 (13) - 1} - 1 = 5 \sqrt{26 - 1} - 1 = 5 \sqrt{25} - 1 = 5 \cdot 5 - 1 = 25 - 1 = 24 \end{aligned}$
::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

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。

Summary
::摘要

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。

Review
::回顾

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

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

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

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

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

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

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

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

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

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

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

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

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

Explore More
::探索更多

1. The legs of a right triangle measure 3 and $\begin{aligned} 2 \sqrt{x} \end{aligned}$ . 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?

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?

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?

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?

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。

6. The motion of an object is modeled by the function $\begin{aligned} s = 2.3 t^{4}, \end{aligned}$ where $\begin{aligned} t \end{aligned}$ is the time (measured in seconds) and $\begin{aligned} s \end{aligned}$ is the position of this object (measured in meters). Find the time when $\begin{aligned} s = \end{aligned}$ 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。

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

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

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 :

References
::参考参考资料

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”

Section outline

Solving Radical Equations ::解决激进等号

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

Example 1 ::例1

WARNING ::警告

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Summary ::摘要

Review ::回顾

Explore More ::探索更多

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

PLIX ::PLIX

References ::参考参考资料

Solving Radical Equations
::解决激进等号

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

Example 1
::例1

WARNING
::警告

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Summary
::摘要

Review
::回顾

Explore More
::探索更多

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

PLIX
::PLIX

References
::参考参考资料