课程： 4.6 二次曲线公式和复杂总和

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二次曲线公式和复杂总和

You probably remember when you first learned to use the quadratic formula in algebra and you ended up with a negative number under the root symbol. Chances are good that your instructor simply said something like "If you get a negative number under the root, there are no real answers, since there is no such thing as the root of a negative!"
::您可能还记得当您第一次学会使用代数中的二次方程式时, 最后在根符号下出现了负数。您的导师说“如果你在根下面有一个负数, 就不会有真正的答案, 因为没有负数的根! ”

Now that you are familiar with , you can probably see that although it would have been an easy assumption at the time that "no real answers" just meant "no answers at all," that isn't true. "No real answers" may well mean that there ARE some "unreal" or imaginary answers.
::既然你已经很熟悉了, 你也许可以看到,虽然在当时,“没有真正的答案”只是“根本没有答案”的意思,但这个假设很简单。“没有真正的答案”很可能意味着存在一些“不真实”或想象中的答案。

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

The Quadratic Formula and the Discriminant
::二次曲线公式和辩驳公式

If ax ² + bx + c = 0
::如果xx2+bx+c=0,则x2+bx+c=0

then $\begin{aligned} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}$
::然后 xbb2 - 4ac2a

Recall that b ² - 4 ac is called the discriminant.
::记得B2 -4ac被称作异见分子

If b ² - 4 ac > 0 then there are two unequal real solutions.
::如果b2 - 4ac > 0,则有两种不平等的实际解决办法。

If b ² - 4 ac = 0 then there are two equal real solutions.
::如果b2 - 4ac = 0,则有两种等效的实际解决办法。

If b ² - 4 ac < 0 then there are two unequal complex solutions.
::如果b2 - 4ac < 0,则有两种不平等的复杂解决办法。

Sums and Differences of Complex Numbers
::复杂数字的总和和差异

When adding (or subtracting) two or more the fastest method is to add (or subtract) the real components to obtain the sum of the real numbers, and then separately add (or subtract) the imaginary coefficients to obtain the sum of the imaginary numbers, e.g.:
::当添加(或减去)两个或两个以上时,最快的方法是添加(或减去)实际组成部分,以获得实际数字的总和,然后单独添加(或减去)想象系数,以获得想象数字的总和,例如:

( a + bi ) + ( c + di ) = [ a + c ] + [ b + d ] i
:a) + b) + (c+ di) = [a+ c] + [b+d] i

Examples
::实例

Example 1
::例1

Combine using addition or subtraction.
::使用加法或减法组合。

$\begin{aligned} (5 + 3 i) + (6 - 8 i) \end{aligned}$
:5+3i)+(6-8i)

Applying the commutative property: $\begin{aligned} (5 + 6) + (3 i + - 8 i) = 11 - 5 i \end{aligned}$
::应用通量属性5+6)+(3i8i)=11-5i

$\begin{aligned} (3 - 2 i) - (2 - 4 i) \end{aligned}$
:3-2i)-(2-4i)

Distribute the negative: $\begin{aligned} (3 - 2 i) + (- 2 + 4 i) \end{aligned}$ , then apply the commutative property: $\begin{aligned} (3 - 2) + (- 2 i + 4 i) = 1 + 2 i \end{aligned}$
:3-2i)+(-2+4i),然后应用通量属性3-2)+(-2)+(-2i+4i)=1+2i

$\begin{aligned} (6) + (4 - 3 i) \end{aligned}$
:6)+(4-3i)

The imaginary coefficient in the first term is 0 , so applying commutative property gives: $\begin{aligned} (6 + 4) + (0 i - 3 i) = 10 - 3 i \end{aligned}$
::第一个学期的想象系数为0,因此应用通融性财产可以产生: (6+4)+(0i-3i)=10-3i

Example 2
::例2

Given: $\begin{aligned} x^{2} + 4 x + 6 = 0 \end{aligned}$ .
::给出: x2+4x+6=0。

$\begin{aligned} x^{2} + 4 x + 6 = 0 \end{aligned}$
::x2+4x+6=0

$\begin{aligned} a = 1, b = 4, c = 6 \end{aligned}$
::a=1,b=4,c=6

$\begin{aligned} x = \frac{- 4 \pm \sqrt{(4)^{2} - (4) (1) (6)}}{2 (1)} \end{aligned}$
::x4(4)2-(4)(1)(6)(2)(1)

Use the discriminant to predict the nature of the roots.
::利用歧见来预测根的本质

Since $\begin{aligned} b^{2} - 4 a c = - 8 \end{aligned}$ , there will be 2 complex solutions (no real solutions)
::由于b2 - 4ac8,将有两个复杂的解决方案(没有真正的解决方案)

$\begin{aligned} x = \frac{- 4 \pm \sqrt{- 8}}{2} \end{aligned}$
::482

Use the quadratic formula to solve and identify the roots.
::使用二次公式解析和识别根。

$\begin{aligned} x = \frac{- 4 \pm 2 i \sqrt{2}}{2} \end{aligned}$
::x42i22

Express the roots as complex numbers in standard form.
::以标准形式将根源作为复杂数字表达出来。

$\begin{aligned} x = - 2 \pm i \sqrt{2} \end{aligned}$
::x2i2

Example 3
::例3

(Graphing calculator exercise)
:绘图计算器练习)

A graphing calculator can perform operations with complex numbers. Press mode. Scroll down and select $\begin{aligned} a + b i \end{aligned}$ . Press Quit. Now the calculator is able to perform operations with complex numbers in a + bi form. When the calculator is in complex number mode, be sure to use parenthesis to group the parts of the complex numbers.
::图形计算器可以执行复杂数字的操作。按模式。向下滚动和选择a+bi 。按 Quit 。现在计算器可以以 + 双形执行复杂数字的操作。当计算器处于复杂数字模式时, 请使用括号来组合复杂数字的各个部分。

Add or subtract the complex numbers using a graphing calculator:
::使用图形化计算器添加或减去复合数字:

$\begin{aligned} (4 - 5 i) - (3 + 2 i) \end{aligned}$
:4-5i)-(3+2i)

$\begin{aligned} 1 - 7 i \end{aligned}$
::1-7i 1-7i

$\begin{aligned} (3 - 7 i) + (2 + i) \end{aligned}$
:3-7i)+(2+1)

$\begin{aligned} 5 - 6 i \end{aligned}$
::5-66i

$\begin{aligned} (- 2 i) + (2 + 6 i) \end{aligned}$
:-2i)+(2+6i)

$\begin{aligned} 2 + 4 i \end{aligned}$
::2+4i 2+4i

$\begin{aligned} (16 - \sqrt{- 64}) + (7 - \sqrt{- 81}) \end{aligned}$

$\begin{aligned} (16 - 8 i) + (7 - 9 i) \end{aligned}$ Simplify the roots in terms of i
:16-8i)+(7-9i)简化i的根

$\begin{aligned} 16 - 8 i + 7 - 9 i \end{aligned}$ Distribute the negative
::16-8i+7-9i 分配负

$\begin{aligned} 23 - 17 i \end{aligned}$ Collect like terms and simplify
::23-17i 收集类似术语和简化

Example 4
::例4

Subtract the complex numbers.
::减去复杂的数字。

$\begin{aligned} (9 + 7 i) - (6 + 3 i) \end{aligned}$
:9+7i)-(6+3i)

$\begin{aligned} (9 + 7 i) + (- 6 - 3 i) \end{aligned}$ Distribute the negative
:9+7i)+(-6-3i) 分配负

$\begin{aligned} (9 + - 6) + (7 i + - 3 i) \end{aligned}$ Group the real part and the imaginary part of each
:96)+(7i3i) 组合每部分的真实部分和想象部分

$\begin{aligned} 3 + 4 i \end{aligned}$ Combine like terms
::3+4i 类似条件组合

$\begin{aligned} (18 + \sqrt{- 81}) - (18 - \sqrt{- 64}) \end{aligned}$

$\begin{aligned} (18 + 9 i) - (18 - 8 i) \end{aligned}$ Simplify the roots
:18+9i)-(18-8i) 简化根

$\begin{aligned} 18 + 9 i - 18 + 8 i \end{aligned}$ Distribute the negative
::18+9i-18+8i

$\begin{aligned} 18 - 18 + 9 i + 8 i \end{aligned}$ Group real and imaginary parts
::18 - 18+9i+8i 组真实和想象部件

$\begin{aligned} 17 i \end{aligned}$ Simplify
::17i 简化

Example 5
::例5

Solve the equations and express them as complex numbers.
::解开方程式,以复杂数字表示。

$\begin{aligned} 18 x^{2} - 2 x + 24 = 0 \end{aligned}$
::18x2-2x+24=0

$\begin{aligned} 9 x^{2} - x + 12 = 0 \end{aligned}$ Divide both sides by 2
::9x2-xx+12=0 将两边除以 2

$\begin{aligned} A = 9 | B = - 1 | C = 12 \end{aligned}$ Identify A, B, and C using standard form: $\begin{aligned} A x^{2} + B x + C = 0 \end{aligned}$
::A=9B=1C=12 使用标准表格识别A、B和C:Ax2+Bx+C=0

$\begin{aligned} \frac{1 \pm \sqrt{1 - 4 (9) (12)}}{2 (9)} \end{aligned}$ Substitute the terms into the quadratic formula $\begin{aligned} \frac{- B \pm \sqrt{B^{2} - 4 A C}}{2 A} \end{aligned}$
::11-4(9)(9)(12.2)/9 将术语替换为四方公式-BB2-4AC2A

$\begin{aligned} x = \frac{1 \pm i \sqrt{431}}{18} \end{aligned}$ By the quadratic formula
::x=1i43118 以二次方

$\begin{aligned} 12 \frac{4}{5} x^{2} = 14 \frac{2}{5} x - 11 \frac{1}{5} \end{aligned}$
::1245x2=1425x-1115

$\begin{aligned} \frac{64}{5} x^{2} - \frac{72}{5} x + \frac{56}{5} = 0 \end{aligned}$ Convert to improper fractions
::645x2- 725x+565=0 转换成不当分数

$\begin{aligned} 64 x^{2} - 72 x + 56 = 0 \end{aligned}$ Multiply both sides by 5
::64x2-72x+56=0 将两边乘以 5

$\begin{aligned} 8 x^{2} - 9 x + 7 = 0 \end{aligned}$ Divide both sides by 8
::8x2-9x+7=0 双方除以8

$\begin{aligned} A = 8 | B = - 9 | C = 7 \end{aligned}$ Extract values for the quadratic formula
::A=8B9C=7 二次公式的提取值

$\begin{aligned} x = \frac{9 \pm i \sqrt{143}}{16} \end{aligned}$ By the quadratic formula
::x=9i14316 以二次公式

Review
::回顾

Add the complex numbers.
::添加复数。

$\begin{aligned} (4 + 4 i) + (3 + 6 i) \end{aligned}$
:4+4i)+(3+6i)

$\begin{aligned} (4 + 8 i) + (8 + 8 i) \end{aligned}$
:4+8i)+(8+8i)
$\begin{aligned} 20 \sqrt{- 100} + \sqrt{- 9} \end{aligned}$
$\begin{aligned} 3 \sqrt{- 121} + 15 \sqrt{- 81} \end{aligned}$
$\begin{aligned} - 25 \sqrt{- 9} + 23 \sqrt{- 196} \end{aligned}$

Subtract the complex numbers.
::减去复杂的数字。

$\begin{aligned} (9 + 8 i) - (8 + 5 i) \end{aligned}$
:9+8i)-(8+5i)

$\begin{aligned} (6 - 8 i) - (12 + 9 i) \end{aligned}$
:6-8i)-(12+9i)
$\begin{aligned} - 5 \sqrt{- 49} - 16 \sqrt{- 1} \end{aligned}$
$\begin{aligned} 22 \sqrt{- 144} - 9 \sqrt{- 4} \end{aligned}$
$\begin{aligned} 27 \sqrt{- 196} - \sqrt{- 1} \end{aligned}$

Solve each equation and express the result as a complex number.
::解决每个方程式,以复数表示结果。

$\begin{aligned} x^{2} - 4 x + 5 = 0 \end{aligned}$
::x2 - 4x+5=0

$\begin{aligned} 3 x^{2} - 6 x + 15 = 0 \end{aligned}$
::3x2-6x+15=0

$\begin{aligned} 8 x^{2} - 5 x + 11 = 0 \end{aligned}$
::8x2 - 5x+11=0

$\begin{aligned} x^{2} - 2 x / 5 + 1 / 5 = 0 \end{aligned}$
::x2-2x/5+1/5=0

$\begin{aligned} - 36 x^{2} - 18 x + 6 = 27 \end{aligned}$
::- 36x2-18x+6=27

When the sum of -4 + 8i and 2 - 9i is graphed, in which quadrant does it lie?
::当 -4 + 8i 和 2 - 9i 的总和被图形显示时, 它在哪个象限中 ?

If $\begin{aligned} z_{1} = - 3 + 2 i \end{aligned}$ and $\begin{aligned} z_{2} = 4 - 3 i \end{aligned}$ , in which quadrant does the graph of $\begin{aligned} (z_{2} - z_{1}) \end{aligned}$ lie?
::如果z13+2i和z2=4-3i,(z2-z1)的图在哪个象限内?

On a graph, if point A represents $\begin{aligned} 2 - 3 i \end{aligned}$ and point B represents $\begin{aligned} - 2 - 5 i \end{aligned}$ , which quadrant contains $\begin{aligned} A - B \end{aligned}$ ?
::在图表中,如果A点代表2-3i,B点代表-2-5i,哪个象限包含A-B?

Find the sum of $\begin{aligned} (- 3 + 4 i) \end{aligned}$ and $\begin{aligned} (- 4 - 7 i) \end{aligned}$ and graph the result
::查找(-3+4i)和(-4-7i)的总和并绘制结果图

Graph the difference of $\begin{aligned} (4 + 7 i) \end{aligned}$ and $\begin{aligned} (2 - 3 i) \end{aligned}$
:4+7i)和(2-2-3i)之差图

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。

章节大纲

常规

二次曲线公式和复杂总和

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

The Quadratic Formula and the Discriminant ::二次曲线公式和辩驳公式

Sums and Differences of Complex Numbers ::复杂数字的总和和差异

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)