二次函数根

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Lesson Objectives
::经验教训目标

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• Find the zeros of a quadratic given the equation in factored form .
  ::以乘数形式查找给定方程的二次方形的零。
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• Understand the .
  ::理解这一点。
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• Solve real-world problems involving a quadratic.  
  ::解决现实世界的问题 涉及到一个二次体。

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Introduction: Fireworks
::导言:烟花

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One characteristic of quadratic functions not discussed in the previous section, Characteristics of Quadratic Functions, was the $\begin{align*}x\text{-}\end{align*}$ intercepts .   T he x-intercept(s) of a function can be found at the point where the $\begin{align*}y\text{-}\end{align*}$ coordinate is 0 , giving an equation of the form   $\begin{align*}0 = ax^2 + bx + c.\end{align*}$  T his section will explore solving this type of equation. 
::前一节未讨论的二次函数的一个特征,即“二次函数”的特征是 X 界面。函数的 x 界面可以在 Y 坐标为 0 的点找到,给出了表0= ax2+bx+c的方程式方程式。本节将探讨如何解决这种方程式。

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A firework is launched from a platform 29 feet off the ground with an initial velocity of 112 ft/s. The height of the firework can be modeled as a function of time using the function  $\begin{align*}h(x) = -16x^2 + 112x + 29.\end{align*}$
::从离地29英尺的平台上发射烟花,初始速度为112英尺/秒。烟花的高度可以用h(x)16x2+112x+29的函数作为时间函数模型。

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The firework must go off at 200 feet to ensure the safety of the onlookers. At what time (h ow many seconds after launch)  should the firework go off?
::为了保证旁观者的安全,烟花必须在200英尺的高度起火。 烟花何时起火(发射后几秒钟)?

 

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Activity 1: Zero Product Principle
::活动1:零产品原则

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Before  answering this question,  you should quickly review the zero product principle. What does it mean if the product of two numbers is zero? Use the interactive below to multiply different integers by zero.
::在回答这个问题之前, 您应该快速检查零产品原则 。 如果两个数字的产值为零, 那意味着什么 ? 使用下面的交互效果将不同的整数乘以零 。

 

 

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Example
::示例示例示例示例

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Solve  $\begin{align*}(x + 5)(2x - 3) = 0\end{align*}$
::解决 (x+5)(2x-3)=0

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The two binomial factors being multiplied in this equation are  $\begin{align*}(x + 5)\end{align*}$  and  $\begin{align*}(2x - 3).\end{align*}$  If one of these factors equals 0, the whole expression will equal zero. Find the values of $\begin{align*}x\end{align*}$   that will make each factor equal to zero.
::在此方程中乘以的两个二进制系数是 (x+5) 和 (2x- 3) 。 如果其中的一个系数等于 0, 则整个表达式等于 0。 查找 x 的值, 使每个系数等于 0 。

$$\begin{align*}\large{x + 5} &\: \large{=} \:0\\ \large{-5} &\:\:\: \large{-5} \\ \hline \large{x} \:\:\:\:\:\:\:& \large{= -5}\end{align*}$$

$$\begin{align*}\large{2x - 3} \:& \large{=}\: 0\\ +\large{3} &\:\:\: +\large{3}\\ \hline \large{2x} \:\:\:\:\:\:\:& \large{=} \:\:3 \\ \div\large{2} \:\:\:\:\:\:\:\:\:& \:\:\: \div\large{2} \\ \hline \large{x} \:\:\:\:\:\:\:&\large{=} \:\:1.5 \end{align*}$$

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Answer : The solutions are -5 and 1.5. Both of these solutions will make the polynomial equal to zero.
::回答:解决办法是-5和1.5。这两种解决办法都将使多元性等于零。

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T his property can be extended to find the roots of polynomial expressions. Q uadratic functions in factored form help find the $\begin{align*}x\text{-}\end{align*}$ intercepts of the function.
::此属性可以扩展以查找多式表达式的根部。 参数形式的二次函数有助于查找函数的 X 界面 。

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Example
::示例示例示例示例

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The height as a function of distance traveled from the character’s position along the $\begin{align*}x\text{-}\end{align*}$ axis can be modeled by the function  $\begin{align*}f(x)= -0.05(x+1)(x-39).\end{align*}$  Find the $\begin{align*}x\text{-}\end{align*}$ intercepts of this function and state what they mean in the context of the problem. 
::函数 f( x)\\\\ 0.05( x+1)( x- 39) 可以以函数 f( x)\\\ \\ \ \ \ \ x- 39 ) 模拟从字符在 X 轴位置上移动的距离函数的高度。 查找此函数的 x 界面, 并指明在问题背景下它们的含义 。

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The $\begin{align*}x\text{-}\end{align*}$ intercept of a function is the input value for which  $\begin{align*}f(x) = 0.\end{align*}$
::函数的 x 界面是 f( x) =0 的输入值。

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$$\begin{align*}0 = -0.05(x+1)(x-39)\end{align*}$$
::00.05(x+1)(x-39)

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We can use the zero product principle to solve for the solutions to this equation.
::我们可以使用零产品原则来解决这一方程式的解决方案。

$$\begin{align*}\large{0\neq -0.05}\end{align*}$$

$$\begin{align*}\large{x}+\large{1}\:&\large{=}\:\:0\\ \large{-1}&\:\:\: \large{-1}\\ \hline \large{x} \:\:\:\:\:\:\:\:&\large{= -1}\end{align*}$$

$$\begin{align*}\large{x - 39} \:&\large{=} \:0\\ +\large{39}&\:\:+\large{39}\\ \hline \large{x}\:\:\:\:\:\:\:\:\:\:&\large{=}\:39\end{align*}$$

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Answer: The $\begin{align*}x\text{-}\end{align*}$ intercepts of the function are  $\begin{align*}(-1,0)\end{align*}$  and  $\begin{align*}(39,0).\end{align*}$
::答复:该函数的x拦截号为(-1,0)和(39,0)。

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The $\begin{align*}x\end{align*}$  values which make a polynomial equal to zero are also called the roots . When the context of a function is applied, they are often referred to as the zeros of the function.  Based on  the $\begin{align*}x\text{-}\end{align*}$ intercepts, the character will be touching the ground after traveling -1 and 39 meters . Ignore the negative $\begin{align*}x\text{-}\end{align*}$ intercept because it happens before the domain of [0,39]. After traveling 39 meters , the character will land.
::使多面等于零的 x 值也称为根。 当应用函数的上下文时, 它们通常被称为函数的零。 根据 x 拦截, 字符在旅行 - 1 和 39 公尺后将触碰地面。 忽略负的 x 拦截, 因为它发生在[ 039] 域之前。 旅行39 公尺后, 字符将着陆 。

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In the expression  $\begin{align*}-0.05(x+1)(x-39),\end{align*}$  each factor is important to the context of the scenario. The factor $\begin{align*}(x + 1)\end{align*}$  represents the starting position if the character was starting at ground level. The factor $\begin{align*}(x - 39)\end{align*}$  represents the position along the $\begin{align*}x\text{-}\end{align*}$ axis where the character will land. Finally, the -0.05 represents the vertical shrink/ stretch of the arc of the character's jump. The farther from 0, the higher, the character will jump.
::在- 0.05 (x+1)(x- 39) 表达式中,每个系数对于设想情景的背景都很重要。 系数( x+1) 表示字符从地面开始时的起始位置。 系数( x- 39) 表示字符会降落的 x 轴沿线的位置。 最后, 系数 - 0.05 表示字符跳动弧的垂直缩缩进/伸展。 越远, 越高, 字符会跳动 。

 

 

 

 

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Activity 2: Using Factoring
::活动2:利用保理

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To answer the question in the introduction,  you need the   Z ero P roduct P rinciple .
::为了回答导言中的问题,你需要 " 零产品原则 " 。

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Example
::示例示例示例示例

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A firework is launched from a platform 29 feet off the ground with an initial velocity of 112 ft/s. The height of the firework can be modeled as a function of time using the function  $\begin{align*}h(x) = -16x^2 + 112x + 29.\end{align*}$  To ensure the safety of the onlookers, the firework must go off at 200 feet. At  how many seconds after launch  should the firework go off?
::从离地29英尺的平台上发射烟花,最初速度为112英尺/秒。烟花的高度可以用h(x)16x2+112x+29的功能作为时间函数模型。为了确保旁观者的安全,烟花必须在200英尺的距离内熄灭。烟花在发射后几秒钟内应熄灭?

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F irst, use a graph to obtain and make sense of the answer, and then consider how  to obtain it algebraically. Use the graph below to graph the function and explore different input-output pairs.
::首先,使用一个图表来获取和理解答案, 然后考虑如何从代数角度获得答案。 使用下图来绘制函数图, 并探索不同的输入- 输出对 。

 

 

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To solve this problem algebraically, 200 can be substituted 200 for the height in  $\begin{align*}h(x).\end{align*}$  
::为了从代数角度解决这个问题,200个以h(x)为单位的高度可以取代200个。

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$$\begin{align*}200 = -16x^2 +112x + 29\end{align*}$$
::20016x2+112x+29

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M ove the 200 to the other side of the equation, to take advantage of the Z ero P roduct P rinciple.
::将200号移到方程的另一边,以利用零产品原则。

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$$\begin{align*}200 &= -16x^2 +112x + 29 \\ -200 &\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-200\\ \hline 0 &= -16x^2 +112x - 171\end{align*}$$
::20016x2+112x+29-200-2000}16x2+112x-171

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U se decomposition to factor this expression:
::使用分解来乘以此表达式 :

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1. Separate the x- term into the sum of two terms that add up to the original x-term but multiply to the product of the  $\begin{align*}x^2\end{align*}$  term and the constant
::1. 将x-期分为两个条件之和的x-期,两个条件加起来等于原来的x-期,但乘以x2-期和常数的产物

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The product of  $\begin{align*}-16x^2\end{align*}$  and the constant  $\begin{align*}-171\end{align*}$  is  $\begin{align*}2,736x^2.\end{align*}$  The number  $\begin{align*}2,736\end{align*}$  has a lot of factors, but  $\begin{align*}36x\end{align*}$  and  $\begin{align*}76x\end{align*}$  multiply to  $\begin{align*}2,736x^2\end{align*}$  and add to  $\begin{align*}112x.\end{align*}$
::16x2和恒定-171的产物为2,736x2。 数字2,736有很多因素,但36x和76x乘以2,736x2,加上112x。

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$$\begin{align*}0= -16x^2 +36x + 76x- 171\end{align*}$$
::016x2+36x+76x-171

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2. Take the greatest common factor out of the first two terms and the last two terms.
::2. 在前两个任期和最后两个任期中采用最大的共同因素。

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$$\begin{align*}0&= -16x^2 +36x + 76x- 171\\ 0&=-4x(4x - 9) + 19(4x- 9)\\ \end{align*}$$
::016x2+36x+76x-1710=4x(4x-9)+19(4x-9)

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3. Take the greatest common factor out of the result from the previous step.
::3. 从前一步骤的结果中取出最大的共同因素。

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$$\begin{align*}0&=-4x(4x - 9) + 19(4x- 9)\\ 0&=(-4x+19)(4x-9)\end{align*}$$
::04x(4x-9)+19(4x-9)+19(4x-9)=(-4x+19)(4x-9)

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The  answers are the values of $\begin{align*}x\end{align*}$  that make each factor equal to zero.
::答案是x的值,使每个因素等于零。

$$\begin{align*}\large{-4x + 19} &\large{= 0}\\ \large{-19}&\:\large{-19}\\ \hline \large{-4x} &\large{= -19}\\ \div\large{-4} \:\:& \:\div\large{-4}\\ \hline \large{x} &\large{= 4.75}\end{align*}$$

$$\begin{align*}\large{4x - 9} &\large{= 0}\\ +\large{9}&\:+\large{9}\\ \hline \large{4x} &\large{= 9}\\ \div\large{4} \:\:& \:\div\large{4}\\ \hline \large{x} &\large{= 2.25}\end{align*}$$

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Answer : The firework will be 200 feet at 2.25 seconds and 4.75 seconds. Does it matter whether the firework goes off at 2.25 seconds and 4.75 seconds?
::回答:烟花将在2.25秒和4.75秒时为200英尺。烟花是否在2.25秒和4.75秒时爆炸,是否重要?

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The solutions represent the x-intercepts to the function  $\begin{align*}h(x)=-16x^2 +112x - 171.\end{align*}$
::解决方案代表函数 h(x)\\\\\\\16x2+112x-171的 x 界面。

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We can find the $\begin{align*}x\text{-}\end{align*}$ intercepts of a quadratic function by factoring the function and using the Z ero P roduct P roperty to solve for $\begin{align*}x.\end{align*}$  Use the interactive below to practice finding the zeros of a function.
::我们能找到四方函数的 X 界面, 方法是将函数乘以系数, 并使用零产品属性来解决 x 。 使用下面的交互功能来练习找到函数的零 。

 

 

 

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Extension: Quadratic Equation Visuals
::扩展名: 二次赤道视觉

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A lgebra tiles are a great tool to visualize solving a quadratic equation . Use the interactive below to explore this.
::代数瓦是可视化解析二次方程的极好工具。 使用下面的交互方式来探索它 。

 

 

 

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Activity 3: The Square Root Method
::活动3: " 平根方法 "

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Another method for solving a quadratic is the square root method. This method  is useful when working with a quadratic of the form $\begin{align*}f(x)=ax^2 + c.\end{align*}$  When using the square root method, you are essentially directly solving for $\begin{align*}x\end{align*}$  using the following steps:
::解决二次方根法是解决二次方根法的另一种方法。在使用 f(x) = ax2+c 窗体的二次方根法时,该方法非常有用。在使用平方根法时,您基本上直接解决 x 使用以下步骤:

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1. Isolate the square.
   ::隔离广场
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2. Take the square root of both sides. 
   ::以双方的平方根。
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3. Solve for $\begin{align*}x.\end{align*}$  
   ::解决x。

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Example
::示例示例示例示例

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$\begin{align*}5x^2 - 7 = 0\end{align*}$
::5x2-7=0

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1. Isolate the square.
::1. 隔离广场。

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The square is   $\begin{align*}x^2.\end{align*}$
::方形为 x2 。

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$$\begin{align*}5x^2 - 7 \:&=\:\: 0\\ +7 & \:\:\:+7\\ \hline 5x^2 \:\:\:\:\:\:\:\:&= \:\:7\\ \div5 \:\:\:\:\:\:\:\:\:\:\:\:& \:\:\:\div5\\ \hline x^2 \:\:\:\:\:\:\:\:&= 1.4\end{align*}$$
::5x2-7=0+7+75x2=7=5=5x2=1.4

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2. Take the square root of both sides.
::2. 以双方的平方根为根。

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$$\begin{align*}\sqrt{x^2}&=\:\:\:\sqrt{1.4}\\ x\:\:&=\pm \sqrt{1.4}\end{align*}$$
::x21.4x1.4

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Recall from Algebra 1 that the opposite operation of squared is the square root. When taking the square root, a plus or minus needs to be included. There are two answers because when a negative number is squared, it becomes positive.
::从代数 1 中回顾, 正方形的相反操作是平方根。 选择平方根时, 需要包含一个正增减。 有两个答案, 因为当负数是正数时, 它就会变成正数 。

• $\begin{align*}\sqrt{1.4}\cdot\sqrt{1.4} = 1.4\end{align*}$  
• $\begin{align*}-\sqrt{1.4}\cdot-\sqrt{1.4} = 1.4\end{align*}$  

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3. Solve for $\begin{align*}x.\end{align*}$  
::3. 解决x.

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We are finished and can be skipped  s ince $\begin{align*}x\end{align*}$  is already isolated on one side. The next example demonstrates how this will not always be the case.
::我们已经完成, 并且可以跳过, 因为 x 已经在一边被孤立了 。 下一个例子说明情况不会总是如此 。

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Answer : The solutions to the equation are  $\begin{align*}+\sqrt{1.4}\end{align*}$  and  $\begin{align*}-\sqrt{1.4}\end{align*}$
::答复:等式的解决方案是1.4和1.4。

 

 

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The square root method can be used to find the $\begin{align*}x\text{-}\end{align*}$ intercepts of a function in vertex form. 
::平方根方法可用于查找顶点形式的函数的 x 界面 。

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Example
::示例示例示例示例

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Jason is a product manager for a home security company. Through focus groups and testing, he has modeled the following function to represent the profit, in thousands, as a function of the price for a new home security camera:  $\begin{align*}f(x) = -(x - 32)^2 + 529.\end{align*}$  Given this function, find the $\begin{align*}x\text{-}\end{align*}$ intercepts and state what they mean in the context of the problem.
::Jason是一家家庭保安公司的产品经理,通过重点小组和测试,他模拟了以下功能,以代表数千美元的利润,作为新的家庭保安摄像头价格的函数:f(x)(x-32)2+529。 有了这一功能,他可以找到 X 接口,并用问题来说明其含义。

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Since the $\begin{align*}x\text{-}\end{align*}$ intercepts can be found by locating the input when $\begin{align*}f(x) = 0,\end{align*}$   begin by  setting the output equal to 0.
::由于 x 界面可以通过 f( x) =0 时定位输入找到, 开始设定输出等于 0 。

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1. Isolate the squared term.
::1. 将平方术语孤立起来。

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T he square is   $\begin{align*}(x-32)^2.\end{align*}$  
::广场是(x-32)。

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$$\begin{align*}0 \:&=\: -(x - 32)^2 + 529\\ -529\:&\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:-529\\ \hline -529 \:&=\: -(x - 32)^2\\ \div-1\:&\:\:\div-1\\ \hline 529 \:&=\:\:\: (x - 32)^2\\\end{align*}$$
::0(x-32)2+529-529-529-529(x-32)211529=(x-32)2

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2. Take the square root of both sides.
::2. 以双方的平方根为根。

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$$\begin{align*}\sqrt{529} \:&=\sqrt{(x - 32)^2}\\ \pm \sqrt{529} \:&=\:\:\:\: (x - 32)\\ ±23 &= \:\:\:\:\:x - 32\end{align*}$$
::529(x-32529=(x-3223=x-32)

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Once the expression which is being squared is isolated,  the  power must be address ed . Additionally, when taking the square root of both sides of an equation, not only will the  exponent cancel, but a plus or minus must be written before the square root of 529. The plus or minus symbol is necessary because the equation has two possible answers. When squaring a number to get 529, both 23 and -23 will equal 529.
::当正方形的表达式被孤立时,必须解决权力问题。此外,在选择方程式两侧的平方根时,不仅推手会取消,而且在529平方根之前必须写出增减。 增减符号是必需的,因为方程式有两个可能的答案。 当用数字对数值进行对齐以获得529, 23和 -23将等于529时, 增减符号将等于529。

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3. Solve for $\begin{align*}x.\end{align*}$  
::3. 解决x.

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The plus-minus represents the two potential values for $\begin{align*}x,\end{align*}$  and as a result, two equations will need to be solved: one with 23 and one with -23.
::加减值代表x的两个潜在值,因此,需要解决两个方程式:一个是23,一个是23。

$$\begin{align*}\large{23} &\large{= x - 32}\\ +\large{32}&\:\:\:\:\:\:\:+\large{32}\\ \hline \large{55} &\large{= x}\end{align*}$$

$$\begin{align*}\large{-23} &\large{= x - 32}\\ +\large{32} &\:\:\:\:\:\:\:+ \large{32}\\ \hline \large{9} &\large{= x}\end{align*}$$

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These solutions represent the $\begin{align*}x\text{-}\end{align*}$ coordinates of the $\begin{align*}x\text{-}\end{align*}$ intercepts.
::这些解决方案代表 x 界面的x 坐标 。

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Answer : The $\begin{align*}x\text{-}\end{align*}$ intercepts are (9,0) and (55,0). The profit from the security camera will be $0 if the price is set to $9 or $55. 
::答复:X接口为(9,0美元)和(55,0美元)。 如果价格定在9美元或55美元,则安全摄像头的利润为0美元。

 

 

 

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Activity 4: Technology
::活动4:技术

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A calculator or mathematical modeling software can be used to solve any quadratic equation. T he intersection of  the functions defined by the left side and the right side of an equation will represent the solutions to the original equation .
::计算器或数学建模软件可用于解析任何二次方程式。由公式左侧和右侧定义的函数的交叉点将代表原始方程式的解决方案。

 

 

 

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Extension: Using Technology to Solve Quadratics
::扩展名: 使用技术解决四边形

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Use the interactive below to explore the use of technology in solving quadratics further .
::利用以下互动方式探索如何利用技术进一步解决二次方块问题。

 

 

 

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Wrap-Up: Review Questions
::总结:审查问题

 

 

 

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播放段落

Extension : Comparing Methods  for Solving Quadratic Equations
::扩展:用于解决赤道等量的比较方法