8.5 使用二次曲线公式解决二次赤道等量

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  使用二次曲线公式解决二次赤道等量

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  A RICE chart allows chemists to keep track of the concentrations of different substances in a chemical reaction ¹ . Cave formation is an example of a chemical reaction in nature. To form a cave, groundwater—which is a weak acid containing carbon dioxide dissolved in water—seeps into the ground and reacts with limestone. Once this mixture reaches the open air of the cave, the reaction reverses and a solid deposit of calcium carbonate is left behind, forming stalagmites and stalactites ² .
  ::RICE 图表使化学家能够追踪化学反应中不同物质的浓度。 洞穴形成是化学反应的一个例子。 要形成洞穴,地下水(水中溶解的含二氧化碳的微弱酸)渗入地面,与石灰石发生反应。一旦这种混合物到达洞穴的露天空气,反应反射和碳酸钙的固体沉积就落在后面,形成炉渣和硫化物2。

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  The result of a RICE chart for a chemical reaction with a weak acid is a quadratic equation , say, for instance,  $\begin{array}{r}{x}^2+{10}^{\text{-}5}x-{10}^{\text{-}6}=0\end{array}$ . It  would be difficult to apply our earlier techniques to this equation . However, we have one more technique for solving quadratic equations in this section.
  ::RICE 微酸化学反应图表的结果是一个二次方程,例如, x2+10-5-5x-10-6=0。在这个方程上很难应用我们先前的技术。然而,我们在本节中还有一种解决二次方程的技术。

  

   

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  Solving Quadratic Equations by the Quadratic Formula
  ::以二次曲线公式解决二次赤道等量

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  The last way to solve a quadratic equation is by using the Quadratic Formula . This formula is derived by for the equation $\begin{array}{r}a{x}^2+bx+c=0\end{array}$ . We will derive the formula here.
  ::解析二次方程的最后方法就是使用二次方程式。 这个公式是用方程式 x2+bx+c=0 来推导的。 我们将在此得出公式 。

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  Investigation: Deriving the Quadratic Formula
  ::调查:提出“四压公式”

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  1. Move the constant to the right side of the equation:  $\begin{array}{r}a{x}^2+bx=\text{-}c\end{array}$ .
  ::1. 将常数移动到方程式的右侧:x2+bx=-c。

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  2. Factor out  $\begin{array}{r}a\end{array}$ from everything on the left side of the equation:  $\begin{array}{r}a\left(x^2+\frac{b}{a}x\right)=\text{-}c\end{array}$ .
  ::2. 从方程左侧的所有东西(a(x2+bax)=-c)中计算出一个值。

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  3. Complete the square using $\begin{array}{r}\frac{b}{a}\end{array}$ :  $\begin{array}{r}{\left(\frac{b}{2}\right)}^2={\left(\frac{\frac{b}{a}}{2}\right)}^2={\left(\frac{b}{2a}\right)}^2=\frac{b^2}{4a^2}\end{array}$ .
  ::3. 使用bab2)2=(ba2)2=(b2a)2=(b2a)2=b24a2,完成广场。

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  4. Add this expression to on the left side. On the right side, you need to multiply it by $\begin{array}{r}a\end{array}$ (to account for the $\begin{array}{r}a\end{array}$ outside the parentheses):  $\begin{array}{r}a\left(x^2+\frac{b}{a}x+\frac{b^2}{4a^2}\right)=\text{-}c+\frac{b^2}{4a}.\end{array}$
  ::4. 向左侧添加此表达式。 在右侧, 您需要将其乘以一个( 用于括号外的计算): a( x2+bax+b24a2) =- c+b24a 。

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  5. Factor the quadratic expression inside the " data-term="Parentheses" role="term" tabindex="0"> parentheses and find the common denominator on  the right-hand side:   $\begin{array}{r}a{\left(x+\frac{b}{2a}\right)}^2=\frac{b^2-4ac}{4a}.\end{array}$
  ::5. 将括号内的二次表达式乘以,在右侧找到共同分母x+b2a)2=b2-4ac4a。

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  6. Divide both sides by $\begin{array}{r}a\end{array}$ :  $\begin{array}{r}{\left(x+\frac{b}{2a}\right)}^2=\frac{b^2-4ac}{4a^2}\end{array}$ .
  ::6. 将双方除以ax+b2a)2=b2-4ac4a2。

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  7. Take the square root of both sides:  $\begin{array}{r}x+\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}\end{array}$ .
  ::7. 以两侧的平方根为平方根: x+b2ab2-4ac2a。

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  8. Subtract $\begin{array}{r}\frac{b}{2a}\end{array}$ from both sides to get $\begin{array}{r}x\end{array}$ by itself:  $\begin{array}{r}x=\frac{\text{-}b\pm \sqrt{b^2-4ac}}{2a}\end{array}$ .
  ::8. 从两边减去b2a以获得x本身:x=-bb2-4ac2a。

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  This formula can be used to solve any quadratic equation.
  ::此公式可用于解析任何二次方程 。

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   How To Using the Quadratic Formula
  ::如何使用二次曲线公式

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  1. Identify    $\begin{array}{r}a,b\end{array}$ , and $\begin{array}{r}c\end{array}$  from $\begin{array}{r}a{x}^2+bx+c=0\end{array}$ .
  ::1. 从x2+bx+c=0中识别a、b和c。

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  2. Substitute the values into the formula and simplify the expression. 
  ::2. 将数值替换为公式并简化表达式。

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  $$\begin{array}{r}x=\frac{\text{-}b\pm \sqrt{b^2-4ac}}{2a}\end{array}$$

   
  ::x=-bb2-4ac2a

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  by Mathispower4u demonstrates how to derive the quadratic formula.  
  ::由 Mathispower4u 演示如何得出二次方程式。

   

   

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  Example 1
  ::例1

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  Solve $\begin{array}{r}2x^2-x-15=0\end{array}$ .
  ::溶解 2x2 -x -15=0 。

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  Solution:  Identify a,b, and c, and then substitute into the formula. 
  ::解决办法:确定a、b和c,然后替换成公式。

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  $$\begin{array}{rl}x& =\frac{1\pm \sqrt{1^2-4(2)(\text{-}15)}}{2(2)}\\ & =\frac{1\pm \sqrt{1+120}}{4}\\ & =\frac{1\pm \sqrt{121}}{4}\\ & =\frac{1\pm 11}{4}\\ & =\frac{12}{4},\text{-}\frac{10}{4}\to 3,\text{-}\frac{5}{2}\end{array}$$

  
  ::x=112-4(2)-152(2)(2)=11+1204=11214=1114=124,-1043,-52

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  This video by CK-12 demonstrates how to use the quadratic formula to solve quadratic equations.
  ::CK-12的这段影片展示了如何使用二次方程式解决二次方程式。

   

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  Example 2
  ::例2

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  Solve $\begin{array}{r}2x^2+5x-15=\text{-}{x}^2+7x+2\end{array}$ using the Quadratic Formula.
  ::使用二次曲线公式解决 2x2+5x-15=-x2+7x+2。

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  Solution:  Get everything onto the left side of the equation.
  ::解决方案:把所有东西都放在方程的左侧。

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  $$\begin{array}{rl}2x^2+5x-15& =\text{-}{x}^2+7x+2\\ 3x^2-2x-17& =0\end{array}$$

  
  ::2x2+5x-15=-x2+7x+23x2-2x-17=0

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  Now, use $\begin{array}{r}a=3,b=\text{-}2,\end{array}$ and $\begin{array}{r}c=\text{-}17\end{array}$ and plug them into the Quadratic Formula.
  ::现在,使用a=3,b=2,和c=17,并插入二次曲线公式。

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  $$\begin{array}{rl}x& =\frac{\text{-}(\text{-}2)\pm \sqrt{(\text{-}2{)}^2-4(3)(\text{-}17)}}{2(3)}\\ & =\frac{2\pm \sqrt{4+204}}{6}\\ & =\frac{2\pm \sqrt{208}}{6}\\ & =\frac{2\pm 4\sqrt{13}}{6}\\ & =\frac{1\pm 2\sqrt{13}}{3}\end{array}$$

  
  ::x=(-2)-(-2)-(-2)-(-2)-2-4(3)(-17)-(3)-(3)-(7)-2(3)=(2)-4+2046=(2)-2086=(2)-2086=(2)-4136=(1)/2133)

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  by Tyler Wallace demonstrates how to use the quadratic formula to solve quadratic equations.  
  ::泰勒·华莱士展示了如何使用二次方程式解决二次方程式。

   

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  Example 3
  ::例3

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  Solve $\begin{array}{r}9x^2-30x-24=0\end{array}$ using the Quadratic Formula.
  ::使用二次曲线公式解决 9x2- 30x- 24=0 。

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  Solution: First make sure one side of the equation is zero. Then find $\begin{array}{r}a,b,\end{array}$ and $\begin{array}{r}c\end{array}$ . $\begin{array}{r}a=9,b=\text{-}30,c=\text{-}24\end{array}$ . Now, plug the values into the formula and solve for $\begin{array}{r}x\end{array}$ .
  ::解决方案 : 首先确保方程式的一面为零。 然后找到 a, b, 和 c. a= 9, b= 30, c= 24。 现在, 将值插入公式中, 并解决 x 。

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  $$\begin{array}{rl}x& =\frac{-(\text{-}30)\pm \sqrt{(\text{-}30{)}^2-4(9)(\text{-}24)}}{2(9)}\\ & =\frac{30\pm \sqrt{900+864}}{18}\\ & =\frac{30\pm \sqrt{1,764}}{18}\\ & =\frac{30\pm 42}{18}\\ & =4,\text{-}\frac{2}{3}\end{array}$$

  
  ::x( 30) ( 30) ( 30) 2- 4( 9) (9)( 242( 9) = 30900+86418= 301, 76418= 30 4218=4, 23

   

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  Example 4
  ::例4

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  The result of a RICE chart for a chemical reaction with a weak acid is a quadratic equation, say for instance,  $\begin{array}{r}{x}^2+{10}^{\text{-}5}x-{10}^{\text{-}6}=0\end{array}$ . Determine the amount of the substance,  x.
  ::RICE 与微酸发生化学反应的 RICE 图表的结果是一个二次方程,例如 x2+10-5x-10-6=0。确定物质的数量, x。

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  Solution:  Note  $\begin{array}{r}a=1,b={10}^{\text{-}5}\end{array}$  and  $\begin{array}{r}c=-{10}^{-6}\end{array}$ . Substituting these values into the quadratic formula yields
  ::溶液:注a=1,b=10-5和c10-6。

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  $$\begin{array}{rl}x& =\frac{\text{-}{10}^{\text{-}5}\pm \sqrt{({10}^{\text{-}5}{)}^2-4(1)(\text{-}{10}^{\text{-}6})}}{2(1)}\\ & =\frac{\text{-}{10}^{\text{-}5}\pm \sqrt{{10}^{\text{-}10}+4\times {10}^{\text{-}6}}}{2}\\ & =\frac{\text{-}{10}^{\text{-}5}\pm \sqrt{4.0001\times {10}^{\text{-}6}}}{2}\\ & =\frac{\text{-}{10}^{\text{-}5}\pm 2.000025\times {10}^{\text{-}3}}{2}\\ & =\frac{\text{-}{10}^{\text{-}5}+2.000025\times {10}^{\text{-}3}}{2}\text{reject negative solution, amount is positive}\\ & =\frac{1.990025\times {10}^{\text{-}3}}{2}\\ & =9.950125\times {10}^{\text{-}4}\end{array}$$

  
  ::x=-10-5-(10-5)2-4-4(1)(-10-6)2(1)=----5-10-10-10+4x10-10-62=-10-5-00.0001x10-62=-5-0.0010-62=--5-2.0025x10-32=----5-5+2.000025x10-32-反转负溶液,正数=1.990025x10-32-32=9.950125x10-4

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     WARNINGS
  ::警告

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  • If  b is negative, remember to change the sign in the 1st term in the formula:  $\begin{array}{r}b=\text{-}2⟶\text{-}b=\text{-}(\text{-}2)=2\end{array}$ .
    ::如果 b 是 负, 则记住更改公式中第一学期的符号 : b=-2 - b=- (-2)= 2。
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  • Divide both terms by 2 a . 
    ::将两个条件除以 2a。
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  • a, b,  and  c represent the coefficients. Do not include the variable with them. 
    ::a、b和c 表示系数,不包括变量。
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  • Simplify the radical 1st, then reduce the fraction .  
    ::简化1号基体,然后降低分数。

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  Summary
  ::摘要

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  • To solve quadratic equations using the quadratic formula, identify the coefficients in the equation and then substitute those values into the formula:  $\begin{array}{r}x=\frac{\text{-}b\pm \sqrt{b^2-4ac}}{2a}\end{array}$ . 
    ::使用二次方程解决二次方程,确定方程中的系数,然后将这些数值替换成公式:x=-bb2-4ac2a。

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  Review
  ::回顾

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  Solve the following equations using the Quadratic Formula:
  ::使用二次曲线公式解决下列方程式:

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  1.  $\begin{array}{r}{x}^2+8x+9=0\end{array}$
  ::1. x2+8x+9=0

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  2.  $\begin{array}{r}\text{-}2x^2+x+5=0\end{array}$
  ::2. -2x2+x+5=0

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  3.   $\begin{array}{r}3x^2=\text{-}4x+5\end{array}$
  ::3. 3x2=-4x+5

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  4.   $\begin{array}{r}{x}^2+5x-150=0\end{array}$
  ::4. x2+5x-150=0

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  5.   $\begin{array}{r}8x^2=2x+3\end{array}$
  ::5. 8x2=2x+3

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  6.   $\begin{array}{r}\text{-}5x^2+18x-24=0\end{array}$
  ::6.5x2+18x-24=0

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  7.  $\begin{array}{r}10x^2+x-2=0\end{array}$
  ::7. 10x2+x-2=0

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  8.  $\begin{array}{r}{x}^2+4=16x\end{array}$
  ::8. x2+4=16x

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  9.  $\begin{array}{r}4x^2+20x+25=0\end{array}$
  ::9. 4x2+20x+25=0

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  10.  $\begin{array}{r}{x}^2-18x-63=0\end{array}$
  ::10. x2-18x-63=0

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  Explore More
  ::探索更多

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  1. The profit on your fundraiser is represented by the quadratic expression  $\begin{array}{r}\text{-}3p^2+200p-3,000\end{array}$ , where p is your price point. What is your break-even point (i.e., the price point at which you will begin to make a profit)? Hint: Set the equation equal to zero.
  ::1. 您筹款人的利润由四边表达式 - 3p2+200p- 3,000表示, p是您的价格点。 您的收支平衡点( 即您开始盈利的价格点) 是什么 ? 提示 : 将方程式设为零 。

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  2. A plane travels at a speed of 160 mph in calm air. Flying with a tailwind, the plane is clocked over a distance of 550 miles. Flying against a headwind, it takes 2 hours longer to complete the return trip. What was the wind velocity? (Round your answer to the nearest tenth.)
  ::2. 飞机飞行速度为160英里,在平静的空气中飞行速度为160英里,飞行时有尾风,飞行时长为550英里,飞行时长为2小时,完成返回旅行需要2小时。 风速是多少? (你对最近的10英里的答复是多少? )

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  3. You want to build a fence around your rectangular yard. The area of your yard is 3,860 ft ² . The width of your neighbor's  yard is 7 feet longer than the length of your yard. What are the dimensions of your yard? Round your answer to the nearest hundredth.  
  ::3. 您想在您的长方形院子周围修建一道栅栏,您的院子面积为3 860平方英尺。您的邻居院子宽度比您的院子长7英尺。您的院子的尺寸是多少? 您的院子的尺寸是多少? 将您的答案绕到最近的100英尺处 。

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  4.  Robert stands on the topmost tier of bleachers in a gymnasium, and throws a basketball out onto the basketball court with a vertical upward velocity of 60 ft/s. The ball is 50 feet above the ground at the moment he releases the ball. When does the ball land?
  ::4. 罗伯特站在健身房最顶端的露天机上,向篮球场投掷篮球,垂直向上速度为60英尺/秒,球在球发球时在地上方50英尺处,球何时落地?

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  5. Find the sum and the product of the two solutions $\begin{array}{r}x=\frac{\text{-}b\pm \sqrt{b^2-4ac}}{2a}\end{array}$ . Note how this relates to how we solve a quadratic equation by factoring.  
  ::5. 找出两种溶液x=-bb2-4ac2a的总和和产品,注意这与我们如何通过保理处理四方方程式有何关系。

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

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  Please see the Appendix. 
  ::请参看附录。

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  References
  ::参考参考资料

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  1. "RICE Chart," last edited March 1, 2017,
  ::1. 2017年3月1日编辑的“RICE图表”,

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  2. " How Caves Form | Caves and Karst | Foundations of the Mendips," by British Geological Survey Natural Environment Research Council, 2017,
  ::2. “洞穴如何形成 洞穴和门底ps的卡斯特基金会”,英国地质调查局自然环境研究理事会,2017年,