课程： 9.4 有平根的平方

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有平根的平方
Running from one base to the next was a speedy 90 feet for Omar. The job of mowing the field, however, probably wasn’t so fast. Just counting the infield, how many square feet of mowing does the groundskeeper do?
::从一个基地跑到另一个基地,对奥马尔来说是90英尺的快速距离。然而,耕种田地的工作可能并不那么快。仅仅计算野外,地基管理员有多少平方英尺的耕种?

In this concept, you will learn how to solve equations using squares and square roots.
::在这个概念中,你将学会如何用方块和平方根解析方程式。

Solving Equations with Square Roots
::用平根解决平方平方

You may already know that squaring a number and taking the square root of a number are opposite operations . If you know one, you can find the other. When working with area and dimensions , the equations most often used are:
::您可能已经知道,对数和对数的平方根进行比对是相反的操作。如果您知道一个, 您可以找到另一个。当使用区域和尺寸时, 最常用的方程式是 :

$\begin{aligned} A = s^{2} \end{aligned}$ when you know the sides and need to find the area.
::A=2 当你了解侧面需要找到区域时 A=2

$\begin{aligned} s = \sqrt{A} \end{aligned}$ when you know the area and want to find the length of the sides.
::s=A 当你知道那个区域想找到两边的长度时

In these equations, both $\begin{aligned} s \end{aligned}$ and $\begin{aligned} A \end{aligned}$ are variables. A variable is simply an unknown quantity represented by a letter. Any letter or symbol can be used in a math sentence as a variable. For example, the equation $\begin{aligned} y = x^{2} \end{aligned}$ says that some number, $\begin{aligned} y \end{aligned}$ , is equal to another number, $\begin{aligned} x \end{aligned}$ , times itself. The equation $\begin{aligned} x = \sqrt{y} \end{aligned}$ is the opposite operation and says some number, $\begin{aligned} x \end{aligned}$ , is equal to the square root of another number, $\begin{aligned} y \end{aligned}$ .
::在这些方程式中, s 和 A 是变量。变量只是字母代表的未知数量。任何字母或符号都可以作为变量在数学句中使用。例如, y=x2 表示, 某数字, y 等于其它数字, x, 乘以它本身。公式 x=y 是相反的操作, 表示某些数字, x 等于另一个数字的平方根, y 。

Here is an example of how to use these equations to solve problems involving squares and square roots.
::这里的例子说明如何利用这些方程式解决涉及平方和平方根的问题。

Solve for $\begin{aligned} y \end{aligned}$ :
::为 y 解决 :

$\begin{aligned} y = 5^{2} \end{aligned}$
::y=52 y=52

First, you know that $\begin{aligned} 5^{2} \end{aligned}$ is the same as $\begin{aligned} 5 \times 5 \end{aligned}$
::首先,你知道52和5x5相同

Next, $\begin{aligned} 5 \times 5 = 25 \end{aligned}$
::下一个, 5x5=25

Then, $\begin{aligned} y = 25 \end{aligned}$
::然后,y=25

Your answer is 25.
::你的答案是25岁

Here’s another example:
::以下是另一个例子:

Solve for $\begin{aligned} x \end{aligned}$ :
::解决 x:

$\begin{aligned} x^{2} = 36 \end{aligned}$
::x2=36

First, you know that $\begin{aligned} x \end{aligned}$ is the unknown variable that you are looking for.
::首先,你知道 x 是您正在寻找的未知变量。

You may also know that in order to find $\begin{aligned} x \end{aligned}$ in any equation , you need to get it on either side of the equal sign, by ITSELF. This means it cannot have anything else attached to it by any other operation, including squares.
::您也可能知道, 要在任何方程式中找到 x, 您需要将其放在 ITISELF 的等同标志的两侧, 也就是说它无法通过任何其他操作, 包括方形来连接它。

Next, in order to isolate $\begin{aligned} x \end{aligned}$ , you must perform the opposite operation. Ask yourself, “WHAT is attached to the $\begin{aligned} x \end{aligned}$ ?” and “HOW is it attached?”
::下一步,为了分离 x, 您必须执行相反的操作。请问您自己 , “ 与 x 连接的是什么 ? ” , “ 与 X 连接的是什么 ? ” 和 “ 与 X 连接的是什么 ? ”

Since a square is attached to the $\begin{aligned} x \end{aligned}$ , the square root of $\begin{aligned} x \end{aligned}$ is the opposite operation.
::由于一个广场附属于x,x的平方根是相反的操作。

Then, remember, that whatever you do to one side of the equation, you must also do to the other side.
::那么,记住,无论你对方程的一边做什么, 你也必须对另一方做什么。

Take the square root of both sides of the equation.
::用方程式两边的平方根

$\begin{aligned} \sqrt{x^{2}} = \sqrt{36} \end{aligned}$
::x2=36

Since you have been given an abstract problem to solve, be sure to include negative roots.
::既然您已经得到了一个抽象的问题需要解决, 请务必包括负面根。

$\begin{aligned} x = \pm 6 \end{aligned}$
::x 6

The answer is $\begin{aligned} \pm 6 \end{aligned}$
::答案是6

Examples
::实例

Example 1
::例1

Earlier, you were given a problem about Omar, who was wondering about how many square feet of grass needed to be mowed.
::更早之前,你被问及Omar, 谁想知道需要修剪多少平方英尺的草地。

One side of the square baseball diamond infield measured 90 feet.
::方形棒球钻场的一面测量了90英尺

First, remember the formula for area of a square.
::首先,记住方块面积的公式。

$\begin{aligned} A = s^{2} \end{aligned}$
::A=s2 阿=2

Next, substitute in what you know.
::下一位,你所知道的替代

$\begin{aligned} A = (90 f t)^{2} \end{aligned}$
::A=(90吨)2

Then solve for $\begin{aligned} A \end{aligned}$ .
::然后解决A。

$\begin{aligned} 90 f t \times 90 f t = 1, 800 s q f t \end{aligned}$
::90 ftx90 ft=1 800 sqft

The answer is 1,800 square feet. Remember that area is measured in squares.
::答案是1800平方英尺记住那个区域是用方形测量的

Example 2
::例2

Solve.
::解决。

$\begin{aligned} x^{2} + 3 = 12 \end{aligned}$
::x2+3=12

First, recognize that you are solving for $\begin{aligned} x \end{aligned}$ and determine the best way to isolate it. In this case there is more than one function attached, a square and a 3 by addition .
::首先, 承认您正在解决 x , 并确定隔离它的最佳方式。在这种情况下, 存在多个附加函数, 一个正方形和一个加3 的函数。

Next, determine which function to remove first from the $\begin{aligned} x \end{aligned}$ . Since you must do the same operation to everything on both sides of the equation, the 3 is the easiest to remove first. You do this by subtraction because it is attached by addition.
::下一步,决定从 x 中先删除哪个函数。既然您必须对等式两侧的所有内容都做相同的操作, 3 是最容易先删除的。您可以通过减法做到这一点, 因为加法是附加的。

$\begin{aligned} x^{2} + 3 - 3 = 12 - 3 \end{aligned}$

::x2+3-3-3=12-3

After subtracting, you have a new equation:
::减去后,您有一个新的方程式 :

$\begin{aligned} x^{2} = 9 \end{aligned}$

::x2=9

Then, all that is left is to take the square root of both sides.
::然后,剩下的就是双方的平方根。

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

::x2=9x3

The answer is $\begin{aligned} \pm 3 \end{aligned}$
::答案是3

Example 3
::例3

Solve for $\begin{aligned} y \end{aligned}$ :
::为 y 解决 :

$\begin{aligned} \sqrt{y - 1} = 8 \end{aligned}$
::y-1=8 y-1=8

First, recognize the two operations attached to the variable - a 1 by subtraction and a square root.
::首先,认列变量所附的两个操作 -- -- 减法1为1,平方根。

Next, determine which operation you can perform first to both sides as a step toward isolating $\begin{aligned} y \end{aligned}$ .
::下一步,决定您可以首先对双方进行哪些操作,作为孤立 y 的一步。

Perform the opposite of a square root by squaring both sides of the equation.
::与平方根对立,对立方程式两侧。

$\begin{aligned} (\sqrt{y - 1})^{2} = 8^{2} \end{aligned}$

:y-1)2=82

This gives you a new equation:
::这给了你一个新的方程式:

$\begin{aligned} y - 1 = 64 \end{aligned}$

::y - 1=64

Then add to remove the 1 that is attached by subtraction.
::然后加上去掉减去后所附的1。

$\begin{aligned} \begin{array}{rcl} y - 1 + 1 & = & 64 + 1 \\ y & = & 64 + 1 \\ y & = & 65 \end{array} \end{aligned}$

::y- 1+1=64+1y=64+1y=64+1y=65

The answer is 65.
::答案是65岁

Example 4
::例4

Solve for $\begin{aligned} x \end{aligned}$ :
::解决 x:

$\begin{aligned} x^{2} = 49 \end{aligned}$
::x2=49

First, take the square root of both sides.
::首先,从双方的平方根开始。

$\begin{aligned} \sqrt{x^{2}} = \sqrt{49} \end{aligned}$
::x2=49

Then, $\begin{aligned} x = \pm 7 \end{aligned}$
::然后, x7

The answer is $\begin{aligned} \pm 7 \end{aligned}$
::答案是++7

Example 5
::例5

Solve for $\begin{aligned} p \end{aligned}$ :
::解决 p:

$\begin{aligned} p^{2} + 5 = 174 \end{aligned}$
::p2+5=174 (p2+5=174)

First, subtract 5 from both sides of the equation.
::首先,从方程两侧减去5。

$\begin{aligned} p^{2} + 5 - 5 = 174 - 5 \end{aligned}$
::p2+5-5=174-5

Next, rewrite the equation:
::下一位, 重写方程式 :

$\begin{aligned} p^{2} = 169 \end{aligned}$
::p2=169 (p2=169)

Then, take the square root of both sides.
::然后,从双方的平方根开始

$\begin{aligned} p = \pm 13 \end{aligned}$
::第13页

The answer is $\begin{aligned} p = \pm 13 \end{aligned}$
::答案是p13

Review
::回顾

Solve each equation.
::解决每个方程式

$\begin{aligned} x^{2} = 9 \end{aligned}$
::x2=9

$\begin{aligned} x^{2} = 49 \end{aligned}$
::x2=49

$\begin{aligned} x^{2} = 100 \end{aligned}$
::x2=100x2=100

$\begin{aligned} x^{2} = 64 \end{aligned}$
::x2=64

$\begin{aligned} x^{2} = 225 \end{aligned}$
::x2=225x2=225

$\begin{aligned} x^{2} = 256 \end{aligned}$
::x2=256xx2=256

$\begin{aligned} x^{2} + 3 = 12 \end{aligned}$
::x2+3=12

$\begin{aligned} x^{2} - 5 = 20 \end{aligned}$
::x2 - 5=20

$\begin{aligned} x^{2} + 3 = 39 \end{aligned}$
::x2+3=39

$\begin{aligned} x^{2} - 4 = 60 \end{aligned}$
::x2 - 4=60

$\begin{aligned} x^{2} + 11 = 92 \end{aligned}$
::x2+11=92

$\begin{aligned} \sqrt{x + 1} = 10 \end{aligned}$
::x+1=10 x+1=10

$\begin{aligned} x^{2} + 5 = 41 \end{aligned}$
::x2+5=41

$\begin{aligned} x^{3} = 8 \end{aligned}$
::x3=8

$\begin{aligned} x^{3} + 4 = 31 \end{aligned}$
::x3+4=31

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

Resources
::资源

章节大纲

常规

有平根的平方

Solving Equations with Square Roots ::用平根解决平方平方

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Example 5 ::例5

Review ::回顾

Review (Answers) ::回顾(答复)

Resources ::资源

Solving Equations with Square Roots
::用平根解决平方平方

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Example 5
::例5

Review
::回顾

Review (Answers)
::回顾(答复)

Resources
::资源