9.14 通过保理公司解决问题

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• 选择节通过保理解决问题

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  通过保理解决问题

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  Solving Problems by Factoring 
  ::通过保理解决问题

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  Now that we know most of the factoring strategies for quadratic polynomials , we can apply these methods to solving real world problems.
  ::既然我们知道四面形多面形动物的 大部分保理策略, 我们可以运用这些方法 来解决现实世界的问题。

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  Real-World Application: Right Triangles 
  ::真实世界应用:右三角

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  One leg of a right triangle is 3 feet longer than the other leg. The hypotenuse is 15 feet. Find the dimensions of the triangle.
  ::右三角形的一条腿比另一条腿长3英尺。 下限为15英尺。 找到三角形的尺寸 。

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  Let $\begin{array}{r}x=\end{array}$ the length of the short leg of the triangle; then the other leg will measure $\begin{array}{r}x+3\end{array}$ .
  ::Let x= 三角形短腿的长度; 然后另一腿将测量 x+3 。

  

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  Use the Pythagorean Theorem : $\begin{array}{r}{a}^2+b^2=c^2\end{array}$ , where $\begin{array}{r}a\end{array}$ and $\begin{array}{r}b\end{array}$ are the lengths of the legs and $\begin{array}{r}c\end{array}$ is the length of the hypotenuse. When we substitute the values from the diagram, we get $\begin{array}{r}{x}^2+(x+3{)}^2={15}^2\end{array}$ .
  ::使用 Pythagorean 理论: a2+b2=c2, 其中 a和 b 是腿的长度, c 是下限的长度。 当我们替换图表中的值时, 我们得到 x2+( x+3) 2=152 。

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  In order to solve this equation , we need to get the polynomial in standard form . We must first distribute, collect like terms and rewrite in the form “polynomial = 0.”
  ::为了解决这个等式, 我们需要以标准形式获得多面体。 我们必须首先以“ 球体=0” 的形式分配、 收集类似条件和重写 。

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  $$\begin{array}{rl}{x}^2+x^2+6x+9& =225\\ 2x^2+6x+9& =225\\ 2x^2+6x-216& =0\end{array}$$

  
  ::x2+x2+6x+9=2252x2+6x+9=2252x2+9=2252x2+6x-216=0

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  Factor out the common monomial : $\begin{array}{r}2(x^2+3x-108)=0\end{array}$
  ::计算出共同单单数 : 2( x2+3x- 108) =0

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  To factor the trinomial inside the " data-term="Parentheses" role="term" tabindex="0"> parentheses , we need two numbers that multiply to -108 and add to 3. It would take a long time to go through all the options, so let’s start by trying some of the bigger factors:
  ::要将括号内的三角因素乘以括号内,我们需要两个乘以-108和加到3的数字。 要通过所有选项需要很长的时间,所以让我们先尝试一些更大的因素:

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  $$\begin{array}{rlrlrl}-108& =-12\cdot 9& & \text{and}& & -12+9=-3\\ -108& =12\cdot (-9)& & \text{and}& & 12+(-9)=3(Correctchoice)\end{array}$$

  
  ::- 129和- 12+93- 108=12(- 9)和12+(- 9)=3(更正选择)

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  We factor the expression as $\begin{array}{r}2(x-9)(x+12)=0\end{array}$ .
  ::我们将表达式乘以2(x-9)(x+12)=0。

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  Set each term equal to zero and solve:
  ::设定每个术语等于零, 并解析 :

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  $$\begin{array}{rlrlrl}& x-9=0& & & & x+12=0\\ & & & \text{or}\\ & \underset{\_}{\underset{\_}{x=9}}& & & & \underset{\_}{\underset{\_}{x=-12}}\end{array}$$

  
  ::x- 9=0x+12=0orx=9__x*12__

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  It makes no sense to have a negative answer for the length of a side of the triangle, so the answer must be $\begin{array}{r}x=9\end{array}$ . That means the short leg is 9 feet and the long leg is 12 feet.
  ::对三角形侧面的长度有一个否定的答案是毫无意义的,所以答案必须是x=9。 这意味着短腿是9英尺,长腿是12英尺。

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  Check: $\begin{array}{r}{9}^2+{12}^2=81+144=225={15}^2\end{array}$ , so the answer checks.
  ::检查: 92+122=81+144=225=152, 所以回答检查 。

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  Real-World Application: Finding Unknown Numbers 
  ::Real- World 应用程序: 查找未知数字

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  The product of two positive numbers is 60. Find the two numbers if one number is 4 more than the other.
  ::两个正数的产物是60, 如果一个数字比另一个数字多4, 请找到这两个数字。

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  Let $\begin{array}{r}x=\end{array}$ one of the numbers; then $\begin{array}{r}x+4\end{array}$ is the other number.
  ::Let x=其中一个数字;然后 x+4 是另一个数字。

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  The product of these two numbers is 60, so we can write the equation $\begin{array}{r}x(x+4)=60\end{array}$ .
  ::这两个数字的产物是60, 所以我们可以写入公式 x( x+4) =60 。

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  In order to solve we must write the polynomial in standard form. Distribute, collect like terms and rewrite:
  ::为了解决问题,我们必须以标准格式书写多边名词。 分配、 收集类似术语和重写 :

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  $$\begin{array}{rl}{x}^2+4x& =60\\ x^2+4x-60& =0\end{array}$$

  
  ::x2+4x=60x2+4x-60=0

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  Factor by finding two numbers that multiply to -60 and add to 4. List some numbers that multiply to -60:
  ::通过发现乘以 -60 乘以 -60 乘以 -60 乘以 4 乘以 两个数字的系数,列出乘以 -60 的数值:

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  $$\begin{array}{rlrlrl}-60& =-4\cdot 15& & \text{and}& & -4+15=11\\ -60& =4\cdot (-15)& & \text{and}& & 4+(-15)=-11\\ -60& =-5\cdot 12& & \text{and}& & -5+12=7\\ -60& =5\cdot (-12)& & \text{and}& & 5+(-12)=-7\\ -60& =-6\cdot 10& & \text{and}& & -6+10=4(Correctchoice)\\ -60& =6\cdot (-10)& & \text{and}& & 6+(-10)=-4\end{array}$$

  
  ::-6+10和6+10=4(修正选择)-60=6(8)(-10)和6+(-10) 4

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  The expression factors as $\begin{array}{r}(x+10)(x-6)=0\end{array}$ .
  ::表达式因数为 (x+10)(x-6)=0。

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  Set each term equal to zero and solve:
  ::设定每个术语等于零, 并解析 :

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  $$\begin{array}{rlrlrl}& x+10=0& & & & x-6=0\\ & & & \text{or}\\ & \underset{\_}{\underset{\_}{x=-10}}& & & & \underset{\_}{\underset{\_}{x=6}}\end{array}$$

  
  ::x+10=0x-6=0orx=10___x=6__

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  Since we are looking for positive numbers, the answer must be $\begin{array}{r}x=6\end{array}$ . One number is 6, and the other number is 10.
  ::由于我们正在寻找正数,答案必须是x=6。 一个数字是6,另一个数字是10。

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  Check: $\begin{array}{r}6\cdot 10=60\end{array}$ , so the answer checks.
  ::检查: 610=60, 所以回答检查 。

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  Real-World Application: Area
  ::实际世界应用:面积

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  A rectangle has sides of length $\begin{array}{r}x+5\end{array}$ and $\begin{array}{r}x-3\end{array}$ . What is $\begin{array}{r}x\end{array}$ if the area of the rectangle is 48?
  ::矩形有长度 x+5 和 x-3 的两边。 如果矩形区域为 48 , 什么是 x ?

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  Make a sketch of this situation:
  ::大致描述一下目前的情况:

  

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  Using the formula Area = length $\begin{array}{r}\times \end{array}$ width, we have $\begin{array}{r}(x+5)(x-3)=48\end{array}$ .
  ::使用公式区域 = 长度 × 宽度,我们有 (x+5)(x-3) = 48。

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  In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and rewrite:
  ::为了解决问题,我们必须以标准格式写下多义。 分配、 收集类似术语和重写 :

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  $$\begin{array}{rl}{x}^2+2x-15& =48\\ x^2+2x-63& =0\end{array}$$

  
  ::x2+2x- 15= 48x2+2x- 63=0

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  Factor by finding two numbers that multiply to -63 and add to 2. List some numbers that multiply to -63:
  ::通过找到乘以 -63 乘以 -63 乘以 -63 乘以 2 乘以 2 的两个数字,列出乘以 -63 乘以 -63 的一些数字:

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  $$\begin{array}{rlrlrl}-63& =-7\cdot 9& & \text{and}& & -7+9=2(Correctchoice)\\ -63& =7\cdot (-9)& & \text{and}& & 7+(-9)=-2\end{array}$$

  
  ::- 79和-7+9=2(更正选择)-63=7(-9)和7+(--9)___2

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  The expression factors as $\begin{array}{r}(x+9)(x-7)=0\end{array}$ .
  ::表达式因数为 (x+9)(x-7)=0。

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  Set each term equal to zero and solve:
  ::设定每个术语等于零, 并解析 :

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  $$\begin{array}{rlrlrl}& x+9=0& & & & x-7=0\\ & & & \text{or}\\ & \underset{\_}{\underset{\_}{x=-9}}& & & & \underset{\_}{\underset{\_}{x=7}}\end{array}$$

  
  ::x+9=0x-7=0orx @%9_x=7__

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  Since we are looking for positive numbers the answer must be $\begin{array}{r}x=7\end{array}$ . So the width is $\begin{array}{r}x-3=4\end{array}$ and the length is $\begin{array}{r}x+5=12\end{array}$ .
  ::由于我们正在寻找正数, 答案必须是 x=7 。 因此, 宽度是 isx- 3=4, 长度是 isx+5=12 。

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  Check: $\begin{array}{r}4\cdot 12=48\end{array}$ , so the answer checks.
  ::检查: 412=48, 所以回答检查 。

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  Example
  ::示例示例示例示例

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  Example 1
  ::例1

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  Consider the rectangle in Example C with sides of length $\begin{array}{r}x+5\end{array}$ and $\begin{array}{r}x-3\end{array}$ . What is $\begin{array}{r}x\end{array}$ if the area of the rectangle is now 20?
  ::如果矩形区域现在是 20 , 则该方形的矩形是 x+5 和 x-3 的两边, 则该方形的矩形是 20 ?

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  Make a sketch of this situation:
  ::大致描述一下目前的情况:

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  Using the formula Area = length $\begin{array}{r}\times \end{array}$ width, we have $\begin{array}{r}(x+5)(x-3)=20\end{array}$ .
  ::使用公式区域 = 长度 × 宽度,我们有 (x+5)(x-3) = 20。

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  In order to solve, we must write the polynomial in standard form. Distribute, collect like terms and rewrite:
  ::为了解决问题,我们必须以标准格式写下多义。 分配、 收集类似术语和重写 :

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  $$\begin{array}{rl}{x}^2+2x-15& =20\\ x^2+2x-35& =0\end{array}$$

  
  ::x2+2x- 15=20x2+2x- 35=0

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  Factor by finding two numbers that multiply to -35 and add to 2. List some numbers that multiply to -35:
  ::通过找到两个乘以-35的乘数,然后加到-2的乘数,列出一些乘以-35的乘数:

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  $$\begin{array}{rlrlrl}-35& =-7\cdot 5& & \text{and}& & -7+5=-2\\ -35& =7\cdot (-5)& & \text{and}& & 7+(-5)=2\end{array}$$

  
  ::- 75和-7+5+52-35=7(-5)和7+(-5-5)=2

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  The expression factors as $\begin{array}{r}(x+7)(x-5)=0\end{array}$ .
  ::表达式系数为 (x+7)(x-5)=0。

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  Set each term equal to zero and solve:
  ::设定每个术语等于零, 并解析 :

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  $$\begin{array}{rlrlrl}& x+7=0& & & & x-5=0\\ & & & \text{or}\\ & \underset{\_}{\underset{\_}{x=-7}}& & & & \underset{\_}{\underset{\_}{x=5}}\end{array}$$

  
  ::x7=0x-5=0orx=0____x=5__

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  Since we are looking for positive numbers the answer must be $\begin{array}{r}x=5\end{array}$ . So the width is $\begin{array}{r}x-3=2\end{array}$ and the length is $\begin{array}{r}x+5=10\end{array}$ .
  ::由于我们正在寻找正数, 答案必须是 x=5 。 因此, 宽度是 isx- 3=2, 长度是 isx+5=10 。

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  Check: $\begin{array}{r}2\cdot 10=20\end{array}$ , so the answer checks.
  ::检查: 210=20, 所以回答检查 。

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  Review 
  ::回顾

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  Solve the following application problems:
  ::解决下列应用问题:

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  1. One leg of a right triangle is 1 feet longer than the other leg. The hypotenuse is 5. Find the dimensions of the right triangle.
     ::右三角的一条腿比另一条腿长1英尺。 下限为 5 。 找到右三角的尺寸 。
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  2. One leg of a right triangle is 7 feet longer than the other leg. The hypotenuse is 13. Find the dimensions of the right triangle.
     ::右三角的一条腿比另一条腿长7英尺。 下限是13。 找到右三角的尺寸 。
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  3. A rectangle has sides of $\begin{array}{r}x+2\end{array}$ and $\begin{array}{r}x-1\end{array}$ . What value of $\begin{array}{r}x\end{array}$ gives an area of 108?
     ::矩形有 x+2 和 x- 1. 的侧面。 x 的值是 108 ?
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  4. A rectangle has sides of $\begin{array}{r}x-1\end{array}$ and $\begin{array}{r}x+1\end{array}$ . What value of $\begin{array}{r}x\end{array}$ gives an area of 120?
     ::矩形的侧面为 x-1 和 x+1 。 x 的值为 120 区域?
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  5. The product of two positive numbers is 120. Find the two numbers if one numbers is 7 more than the other.
     ::两个正数的产物是120 如果一个数字比另一个数字多7个,请找出这两个数字。
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  6. A rectangle has a 50-foot diagonal. What are the dimensions of the rectangle if it is 34 feet longer than it is wide?
     ::矩形有50英尺的对角形。如果矩形长于34英尺,则矩形的尺寸是多少?
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  7. Two positive numbers have a sum of 8, and their product is equal to the larger number plus 10. What are the numbers?
     ::两个正数总和为8,其产品等于较大数字加10。 数字是多少?
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  8. Two positive numbers have a sum of 8, and their product is equal to the smaller number plus 10. What are the numbers?
     ::两个正数总和为8,其产品等于小数加10。 数字是多少?
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  9. Framing Warehouse offers a picture framing service. The cost for framing a picture is made up of two parts: glass costs $1 per square foot and the frame costs $2 per foot. If the frame has to be a square, what size picture can you get framed for $20?
     ::建模仓库提供图片设计服务。 建图成本由两部分组成: 每平方英尺玻璃费用为1美元,每平方英尺框架费用为2美元。 如果框架必须是平方, 那么您可以得到何种尺寸的20美元?

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  Review (Answers)
  ::回顾(答复)

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  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 。