章节大纲

  • Lesson Objectives
    • Factor quadratics by .
      ::以 . xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
    • Know the quadratic formula , and understand how to derive it.
      ::了解二次方程式, 并了解如何获得它。
    • Find the zeros of a quadratic using the quadratic formula.
      ::使用二次方程式查找二次方块的零。
    • Understand that i  is used to represent imaginary numbers.
      ::理解我用来代表想象中的数字。
    • Use the roots of a quadratic to solve real-world problems.
      ::利用二次曲线的根 解决现实世界的问题
    • Understand how to use the discriminant to determine the types of solutions of a quadratic.
      ::了解如何使用歧见者来决定四象体的解决方案类型。
    • Use the discriminant to determine the type of solutions of a quadratic.
      ::使用对立词来确定二次方块的解决方案类型。
    • Factor a quadratic with complex roots.
      ::具有复杂根部的二次方位数 。

    Introduction: Breaking Even
    ::导言:打破平衡

    The profits for a business can be determined by subtracting the costs from the revenue . Suppose the monthly revenue of a business, in thousands, is modeled by the function  R ( x ) = 5.5 x 0.01 x 2 ,  and the monthly cost of manufacturing the product, in thousands, is modeled by  C ( x ) = 300 + 1.5 x ,  where x  is the number of units of the product.  T he profit function can be found by subtracting the cost from the revenue to obtain  P ( x ) = 0.01 x 2 + 4 x 300.  Since break-even means a profit of $0, set  P ( x )  equal to 0. Rather than solving algebraically, consider an alternative approach. Examine the graph of the revenue function and cost function.
    ::企业的利润可以通过从收入中减去成本来确定。假设企业的月收入(千)以R(x)=5.5x-0.01x2的函数为模型,如果企业的月收入(千)以R(x)=5.5x-0.01x2为模型,而产品的月生产成本(千)以C(x)=300+1.5x为模型,其中x是产品单位的数量。利润函数可以通过从收入中减去成本以获得P(x)=0.01x2+4x-300来找到。因为收支平衡意味着相当于0美元的利润,设定P(x)为P(x),而不是解决代数,考虑另一种方法。审查收入功能和成本功能的图表。

    lesson content

    Discussion Questions
    ::讨论问题 讨论问题

    1. How many units of the product need to be sold to break even?
      ::需要出售多少个产品单位才能达到平衡?
    2. Suppose a tax on the product raises the cost function to  C ( x ) = 300 + 2.5 x .  How many units will need to be sold to break even? What problems arise when solving the profit function for 0?
      ::假设产品税将成本函数提高到 C( x) = 300+2.5x。 需要出售多少个单位才能实现平衡? 解决 0 的利润函数会产生什么问题 ?

     


    Activity 1: Completing the Square 
    ::活动1:完成广场

    If the new profit function,  P ( x ) = 0.01 x 2 + 3 x 300 ,  is set equal to zero, the expression  0.01 x 2 + 3 x 300  cannot be factored using any of the methods that have been explored so far. However, this does not mean that the expression cannot be factored and solved. This equation can be solved using a method learned in algebra 1: completing the square. The steps for completing the square are as follows:
    ::如果将新的利润函数P(x)0.01x2+3x-300设定为零,则使用迄今所探讨的任何方法都无法对表达式- 0.01x2+3x-3x-3300进行计算。但这并不意味着该表达式无法计算和解决。这个方程式可以通过在代数1:填平方形中学习的方法加以解决。完成方形的步骤如下:

    1. Divide all terms by the a   coefficient .
      ::将所有条件除以系数。
    2. Move the constant term to the side of the equation that is  opposite to  the variables.
      ::将恒定条件移动到方程式的对面,与变量相对。
    3. Add  ( b 2 ) 2  to both sides of the equation.
      ::将(b2)2添加到等式的两侧。
    4. Complete the square by factoring the equation.
      ::通过乘以方程来填平方形。
    5. Take the square root of both sides and solve.
      ::以双方的平方根 解决。

    Example
    ::示例示例示例示例

    Jane is an architect and was asked to design a home. She is using a 3D modeling program to design the model. Jane wants the length of the house to be six feet more than double the width. If the rectangular base of the house is to cover 685 square feet, what should the dimensions of the house be?
    ::Jane是一名建筑师,被要求设计一个房子。她正在使用3D模型设计模型。 Jane希望房子的长度比宽度高出6英尺。如果房屋的矩形底座要覆盖685平方英尺,那么房屋的尺寸应该有多大呢?

    Before attempting to answer the question, it is a good idea to create a model. A rectangle base that covers  685 square feet refers to the area which can be modeled using the formula Area = length width .   It is given that t he length must be six  feet more than double the width.
    ::在试图解答这个问题之前, 创建模型是一个好主意。 覆盖685平方英尺的矩形基座是指可用公式“ 区域=宽度” 模式建模的区域。 给出的长度必须是宽度的两倍, 六英尺以上。

    • Width =  x  
      ::宽度=x
    • Length =  2 x + 6  
      ::长度=2x+6
    • Area =  ( 2 x + 6 ) x  
      ::面积 = (2x+6) =x

    U se this to write the following equation:
    ::使用此来写入以下方程式 :

    685 = ( 2 x + 6 ) x


    ::685=( 2x+6) x

    A ttempting to solve this equation using the methods  used  thus far would result in the following: 685 = ( 2 x + 6 ) x 685 = 2 x 2 + 6 x 0 = 2 x 2 + 6 x 685


    ::使用迄今为止使用的方法试图解析此方程式, 将产生以下结果 : 685 = (2x+6) x685 = 2x2+6x0= 2x2+6x2+6x- 685

    The expression  2 x 2 + 6 x 685  has no rational factors,  but this equation can still be solved by  completing the square.
    ::表达式 2x2+6x-685 没有合理因素, 但此方程式仍然可以通过完成正方形来解答 。

    1. Divide all terms by the a  coefficient.
    ::1. 将所有条件除以系数。

    0 = 2 x 2 + 6 x 685 ÷ 2 ÷ 2 ÷ 2 ÷ 2 0 = x 2 + 3 x 342.5


    ::0=2x2+6x- 685222222220=x2+3x-34.5

    2. Move the constant term to the side of the equation that is opposite to the variables .
    ::2. 将常数移到方程对面的方程侧面。

    0 = x 2 + 3 x 342.5 + 342.5 + 342.5 342.5 = x 2 + 3 x


    ::0=x2+3x-342.5+342.5+342.5+342.52.5=x2+3x

    3. Add  ( b 2 2 )   to both sides of the equation.
    ::3. 在等式的两边加上(b22)。

    Since the b  coefficient is 3,
    ::因为b系数是3,

    ( b 2 ) 2 = ( 3 2 ) 2 = ( 1.5 ) 2 = 2.25


    :伤心b2)2=(32)2=(1.5)2=2.25

    A dd 2.25 to both sides
    ::向双方增加2.25

    342.5 + 2.25 = x 2 + 3 x + 2.25 344.75 = x 2 + 3 x + 2.25


    ::342.5+2.25=x2+3x+2.25344.75=x2+3x+2.25

    4. Complete the square by factoring the equation.
    ::4. 通过计算方程来完成方形。

    The expression  x 2 + 3 x + 2.25  is a perfect square trinomial and will factor to  ( n x + m ) 2 ,  where n  is the square root of the a-coefficient and m  is half the b-coefficient.
    ::表达式 x2+3x+2.25 是一个完美的正方方形三角, 并将乘以 (nx+m) 2, 其中 n 是系数的正方根, m 是 b- 系数的一半 。

    344.75 = ( x + 1.5 ) 2


    ::344.75=(x+1.5)2

    5. Take the square root of both sides and solve.
    ::5. 解决双方的平方根。

    344.75 = ( x + 1.5 ) 2 ± 344.75 = x + 1.5 1.5 1.5 ± 344.75 1.5 = x


    ::344.75(x+1.5)2344.75=+1.5-1.5-1.5344.75-1.5=x

    Taking the square root will give the two answers:  344.75 1.5   and  344.75 1.5.  Although these solutions are irrational, this does not mean that they are not important to Jane. Re call that an irrational number cannot be expressed as the ratio of two integers , however,=0 it can be estimated.
    ::以平方根来回答两个答案:+344.75-1.5和+344.75-1.5。虽然这些解决办法是不合理的,但这并不意味着它们对Jane不重要,但请注意,不合理数字不能以两个整数的比例表示,但可以估计为0。

    • 344.75 1.5 17.067  feet 
      ::344.75-1.5 177.67英尺
    • 344.75 1.5 20.067   feet
      ::344.75-1.5 2.0067英尺

    In this scenario, the negative value can be disregarded,  so the width of the house should be 17.067 feet. Since the length is 6 more than double the width, the length will be 40.134 feet.
    ::在这种情况下,可以忽略负值,因此房屋宽度应为17.067英尺。由于宽度是6倍以上,长度为40.134英尺。

    Answer : The dimensions of the house should be 17.067 feet by 40.134 feet.
    ::答复:房屋的尺寸应为17.067英尺乘40.134英尺。

     

    Discussion Questions :
    ::讨论问题:

    Could you use  completing the square to  factor the expression  x 2 + 14 x + 39 ?  What challenges would arise? Could you extend this to use completing the square to factor any expression? 
    ::您能否使用完成方块来乘以表达式 x2+14x+39 的乘数? 会出现什么挑战? 您能否将此乘数扩展至填全方块以乘以任何表达式 ?

     


    Activity 2: The Quadratic Formula
    ::活动2:二次曲线公式

    Since the steps  for completing the square are always the same, a formula could be generalized for any quadratic function of the form  f ( x ) = a x 2 + b x + c .  Use the interactive below to explore this.
    ::由于填写正方形的步骤总是相同的,对表f(x)=ax2+bx+c的任何四边函数,公式可以普遍化。 使用下面的交互功能来探索这一点。

    To get a sense of  how the quadratic function works, consider the expression with the numerator separated into two parts, each over the denominator. This is the second to last step in the image above.
    ::要了解二次函数是如何运作的, 请考虑以分子分隔成两个部分的表达式, 每个部分在分母上方。 这是上方图像中第二步到最后一步 。

    b 2 a ± b 2 4 a c 2 a


    ::-b2a-b2-4ac2a

    You  should recognize the term   b 2 a  from the vertex formula from the standard form of a quadratic. This term represents the center of the parabola on the x-axis , also known as the axis of . The term  b 2 4 a c 2 a  is the distance from the center to the x- intercept to the left of the axis of symmetry and to the right of the axis of symmetry. Starting from the center, take  b 2 4 a c 2 a  "steps" in either direction to get to the x- intercepts .
    ::您应该从二次方位标准形式的顶端公式中识别一个术语- b2a。 此术语代表 x 轴上的抛物线中心, 也称为 轴。 b2 - 4ac2a 术语是从中间到对称轴左侧的X 拦截距离, 到对称轴右侧的距离。 从中位开始, 在任何方向上用 b2 - 4ac2a “ 步子” 到 X 截取 。

     


    Activity 3: Understanding Quadratic Solutions
    ::活动3:了解四方解决办法

    Now that  you have two new approaches you can use to solve a quadratic equation take a look at  the question in the introduction. The new cost function will result in the following profit function:
    ::现在你有两种新办法可以用来解决二次方程, 看一下导言中的问题。 新的成本功能将产生以下利润功能 :

    P ( x ) = R ( x ) C ( x ) = ( 5.5 x 0.01 x 2 ) ( 300 + 2.5 x ) = 5.5 x 0.01 x 2 300 2.5 x = 0.01 x 2 + 3 x 300


    ::P(x)=R(x)-C(x)=( 5.5x- 0.01x2)-( 300+2.5x)=5.5x- 0.01x2- 300- 2.5x_ 0.01x2- 300- 0.001x2+3x- 300)

    Example
    ::示例示例示例示例

    U se the quadratic formula to find  the  zeroes of the function f ( x ) = 0.01 x 2 + 3 x 300.
    ::使用二次方程式查找函数 f( x)\\\ 0.01x2+3x- 300 的零。

    Start  by identifying the coefficients.
    ::从确定系数开始。

    • a = 0.01  
      ::0.01美元
    • b = 3  
      ::b=3 =
    • c = 300  
      ::300

    Then s ubstitute these coefficients into the quadratic formula and simplify.
    ::然后将这些系数换成二次公式并简化。

    ( 3 ) ± ( 3 ) 2 4 ( 0.01 ) ( 300 ) 2 ( 0.01 ) 3 ± ( 3 ) 2 4 ( 0.01 ) ( 300 ) 0.02 3 ± 9 4 ( 0.01 ) ( 300 ) 0.02 3 ± 9 12 0.02 3 ± 3 0.02

    At this point in the process, the square root will be addressed. However, the square root of a negative number cannot be taken since no number multiplied by itself can be negative. To deal with situations like this, the letter  i    is used.
    ::在此过程中,正方根将得到解决。 但是, 负数的平方根无法被接受, 因为数字本身不能乘以负数。 要处理这种情况, 请使用字母 i 。

    i = 1


    ::i1

    T he negative can be removed from the square root symbol using i . The letter i  signifies that the number is “imaginary” in that it does not exist within the dimensions being used.
    ::使用 i 可以从平方根符号中删除负数。 字母i 表示数字是“想象的”, 因为它在使用的维度中不存在。

    3 = i 3


    ::3=i3

    For now, t he constan t   i  will be treated as  a variable . However, in ,  the constant   i   and how it differs from a basic variable will be looked at in-depth. R eplace the   1  with i :
    ::目前, 常数 i 将被视为变量。 但是, 常数 i 及其与基本变量的不同之处将会被深入地查看。 将 {% 1 替换为 i :

    3 ± 3 0.02 3 ± i 3 0.02


    ::- 33 - 0.02 - 33 - 0.02

    Once the fraction is fully simplified, write each term in the numerator over the term in the denominator.
    ::分数完全简化后,在分母的分数中,在每个术语的分子中写入每个术语。

    3 0.02 ± i 3 0.02 3 0.02 ± i 3 0.02 150 ± i 3 0.02


    ::- 3 - 0.023 - 0.0230.023 - 0.021503 - 0.02

    Since dividing by -0.02, (or  1 50  as a fraction), is the same as multiplying by -50, the -0.02 can be moved to the numerator as a -50. Additionally,  ± 50   is the same as  ± 50 ,   so  the negative  can be  removed .
    ::由于除以 -0.02 (或- 150 分数) 与乘以 -50 相同,那么 - 0.02 可以移动到分子- 50 。此外, 50 与 50 相同, 负可以删除 。

    150 ± 50 i 3


    ::150503

    Answer 150 + 50 i 3   and  150 50 i 3
    ::答复:150+503和150-503

    What does this mean in the context of a graph? In the introduction, you saw that the x-intercepts of a profit function, which represent the break-even point, are where the revenue function is equal to the cost function. Use the interactive below to examine how the new functions would look when graphed.
    ::这在图表的背景中意味着什么?在导言中,你看到利润函数的 X 拦截点代表平衡点,即收入函数与成本函数相等。使用下面的交互功能来检查图表显示时新函数的外观。

    Discussion Question : Since an x -value of six, satisfies the inequality does this mean that selling 6 objects will yield a profit?
    ::讨论问题:既然X值为6,那么满足了不平等,这是否意味着出售6件物品将产生利润?

     


    Activity 4: The Discriminant
    ::活动4:反对者

    The interactive above shows what an imaginary solution looks like in context. The x-intercepts of a function are imaginary if the function does not cross the x-axis. The type of solutions a quadratic function will have can be determined based on the expression  b 2 4 a c  inside the square root symbol. This expression is called the discriminant.  Answer the questions  below to explore how the discriminant can affect the solutions to a quadratic equation.
    ::上方互动显示一个假想的解决方案在上下文中看起来是什么样。 如果函数不跨过 X 轴, 函数的 X 界面是假想的。 二次函数将具有的解决方案类型可以基于平方根符号中的表达式 b2 - 4ac 来决定。 这个表达式被称为对立。 回答下面的问题, 以探讨对立方程式如何影响四方形的解决方案 。

     

    Discussion Question : Make a conjecture about the number of roots a degree n   polynomial has.
    ::讨论问题: 假设一下根数是多民族的学位。

      

     


    Activity 5: Factoring Prime Quadratics
    ::活动5:保理主要四方

    The quadratic expression   4 x 2 + 1    may appear to  not be factorable at first glance. However, it is possible  to  use the imaginary roots to  write the expression as a product of factors. Recall that a quadratic of form  f ( x ) = a ( x p ) ( x q )  has the solutions p and q.  By obtaining the solutions, you  can reverse engineer the factored form of the equation.  
    ::二次曲线表达式 4x2+1 乍一看可能似乎无法计算。 但是, 可以用想象中的根来写出表达式作为因子的产物。 回顾表f( x)=a( x- p)(x- q) 的二次曲线具有溶液 p和q。 通过获取解决方案, 您可以对公式的系数形式进行反演 。

    Example 
    ::示例示例示例示例

    Factor  4 x 2 + 1
    ::因数 4x2+1

    T he square root method will produce the following solutions:
    ::平方根方法将产生以下解决办法:

    4 x 2 + 1 = 0 1 1 4 x 2 = 1 ÷ 4 ÷ 4 x 2 = 1 4 x 2 = 1 4 x = ± 1 2 i


    ::4x2+1 =0 - 1 - 14x2 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @% 2 @ label: label

    Our p  and q  values are  1 2 i  and 1 2 i ,  respectively. Substituting these into the factored form will result in the following:
    ::我们的p和q价值分别为12i和-12i。

    f ( x ) = 4 ( x 1 2 i ) ( x + 1 2 i )


    :伤心xx)=4(x-12i)(x+12i)

    You  know the  a coefficient  is 4 because the  a coefficient  in the original equation is 4. Given   4 = 2 × 2 ,  distributing a 2 to each factor will cancel any denominators, allowing you to simplify the factors.
    ::4=2x2,每个系数分配一个2,就可以取消任何分母,从而可以简化因素。

    f ( x ) = 4 ( x 1 2 i ) ( x + 1 2 i ) = 2 ( x 1 2 i ) 2 ( x + 1 2 i ) = ( 2 x i ) ( 2 x + i )


    :伤心x)=4(x-12i)(x+12i)(x-12i)(x-12i)(x-12i)=2__(x-12i)(x+12i)(x+12i)(2x-i)(2x+i)

    Answer:   ( 2 x i ) ( 2 x + i )  
    ::答复: (2x-1)(2x+1)

    Answer the questions  below to practice factoring prime quadratic expressions.
    ::回答以下问题,以实践主要二次表达式的保理。

    Discussion Question : Can you use the solutions 150 + 50 i 3  and 150 50 i 3  from the example above to write the factored form of  0.01 x 2 + 3 x 300 ?  
    ::讨论问题:您能否使用上述例子中的150+503和150-50i3的解决方案来写出系数形式为- 0.01x2+3x- 300?

     


    Extension: Newton's Method
    ::扩展名: 牛顿方法

    While  the  quadratic formula is commonly used to identify irrational roots, Isaac Newton devised his own method for calculating irrational roots. Use the interactive below to explore this method.
    ::二次方程式通常用来识别非理性根根, Isaac Newton则自己设计了计算非理性根根的方法。 使用下面的交互式公式来探索这个方法 。

     


    Wrap-Up: Review Questions
    ::总结:审查问题

     


    Extension: Quadratic Formula and Complex Sums
    ::扩展名: 二次曲线公式和复杂总和

       Summary
    ::摘要

    If you cannot factor a quadratic, complete the square to make the quadratic a perfect square trinomial:
    ::如果您不能乘以二次方位,请填写方位,使二次方位成为完美的三重正方位:

    • Divide all the terms by a  
      ::将所有条件除以 a
    • Move the constant to the other side of the equation
      ::将常数移动到方形的另一侧
    • Add ( b 2 ) 2  to both sides
      :伤心b2)2 向双方增加(b2)2
    • Factor the equation
      ::乘以方程
    • Take the square root
      ::取平方根

    Completing the square can be written as an algorithm known as the  quadratic equation where  x = b ± b 2 4 a c 2 a .
    ::完成方形可以写成一种算法, 称为四方形方程式, 其中 xbb2- 4ac2a 。