节: 解决具有复杂数字解决方案的赤道等量 | 8.7 用复杂数字解决方案解决赤道等量

章节大纲

Bryson owns a business that manufactures and sells tires. The revenue from selling tires in the month of July is given by the function $\begin{align*}R(x)=x(200-0.4x)\end{align*}$ , where x is the number of tires sold. Can Bryson's business generate revenue of $30,000 in the month of July?
::Bryson拥有一个制造和销售轮胎的企业,7月份销售轮胎的收入由R(x)=x(200-0.4x)函数提供,而售出的轮胎数量是x。 Bryson的企业在7月份能创收30 000美元吗?

For this scenario we need to deal with complex solutions to a quadratic equation. We discuss those in this section.
::对于这一设想,我们需要处理四方形等式的复杂解决办法,我们在本节中讨论这些解决办法。

The Discriminant
::诽谤者,

The Quadratic Formula is $\begin{align*}x=\tfrac{\text{-}b \pm \sqrt{b^2-4ac}}{2a}\end{align*}$ . The expression under the radical, $\begin{align*}b^2-4ac\end{align*}$ , is called the discriminant. You can use the discriminant to determine the number and type of solutions an equation has.
::二次曲线公式是 x=- bb2- 4ac2a。基下表达式 b2- 4ac, 被称为 driminant。您可以使用 dispriminant 来确定方程式的解决方案的数量和类型。

Investigation: Solving Equations with Different Types of Solutions
::调查:用不同类型解决办法解决等式问题

1. Solve $\begin{align*}x^2-8x-20=0\end{align*}$ using the Quadratic Formula. What is the value of the discriminant?
::1. 使用二次曲线公式解决 x2-8x-20=0。

$\begin{aligned} x & = \frac{8 \pm \sqrt{144}}{2} \\ = \frac{8 \pm 12}{2} \to 10, - 2 \end{aligned}$
$\begin{align*}x &=\frac{8 \pm \sqrt{{\color{green}144}}}{2}\\ &=\frac{8 \pm 12}{2} \rightarrow 10, \text{-}2\end{align*}$
::x=81442=812210,-2

2. Solve $\begin{align*}x^2-8x+16=0\end{align*}$ using the Quadratic Formula. What is the value of the discriminant?
::2. 使用二次曲线公式解决 x2- 8x+16=0。

$\begin{aligned} x & = \frac{8 \pm \sqrt{0}}{2} \\ = \frac{8 \pm 0}{2} \to 4 \end{aligned}$
$\begin{align*}x &=\frac{8 \pm \sqrt{{\color{green}0}}}{2}\\ &=\frac{8 \pm 0}{2} \rightarrow 4\end{align*}$
::x=802=8024

3. Solve $\begin{align*}x^2-8x+20=0\end{align*}$ using the Quadratic Formula. What is the value of the discriminant?
::3. 使用二次曲线公式解决 x2-8x+20=0。

$\begin{aligned} x & = \frac{8 \pm \sqrt{- 16}}{2} \\ = \frac{8 \pm 4 i}{2} \to 4 \pm 2 i \end{aligned}$
$\begin{align*}x &=\frac{8 \pm \sqrt{{\color{green}\text{-}16}}}{2}\\ &=\frac{8 \pm 4i}{2} \rightarrow 4 \pm 2i\end{align*}$
::x=8-162=84i24□2i

4. Look at the values of the discriminants from steps 1 to 3. How do they differ? How does that affect the final answer?
::4. 看看从步骤1到步骤3的反对者的价值,他们有何不同?这对最后答案有何影响?

Determining the Number and Types of Solutions From the Discriminant
::确定来自争议方的解决办法的数目和类型

The discriminant is the expression underneath the radical in the quadratic formula: $\begin{align*}b^2-4ac\end{align*}$ .
::争议是四方形(b2 - 4ac)的基底下方的表达方式:b2 - 4ac。

If $\begin{align*}b^2-4ac = 0\end{align*}$ , then the equation has one real solution, a double root.
::如果 b2 - 4ac=0, 那么方程式有一个真正的解决方案, 一个双根。

If $\begin{align*}b^2-4ac < 0\end{align*}$ , then the equation has no real solutions. It has two complex solutions, and these solutions are complex conjugates of each other.
::如果 b2 - 4ac < 0 , 那么方程式就没有真正的解决方案。它有两个复杂的解决方案, 而这些解决方案是彼此复杂的共同点。

If $\begin{align*}b^2-4ac > 0\end{align*}$ , then the equation has two real solutions.
::如果 b2 - 4ac>0, 那么方程式有两个真正的解决方案。

by Mathispower4u provides four examples of how to describe the type or nature of the solutions to a quadratic equation by calculating the discriminant.
::Mathispower4u 提供了四个例子,说明如何通过计算对立方程式来描述二次方程式解决方案的类型或性质。

Example 1
::例1

Determine the type of solutions $\begin{align*}4x^2-5x+17=0\end{align*}$ has.
::确定 4x2 - 5x+17=0 的解决方案类型。

Solution: Find the discriminant.
::找到这名持不同政见的人

$\begin{aligned} b^{2} - 4 a c & = (- 5)^{2} - 4 (4) (17) \\ = 25 - 272 \\ = - 247 \end{aligned}$
$\begin{align*}b^2-4ac &=(-5)^2-4(4)(17)\\ &=25-272\\ &=\text{-}247\end{align*}$
::b2-4ac=(-5-5)2-4(4)(17)=25-272=247)

This equation has two complex solutions.
::这一方程式有两个复杂的解决办法。

Example 2
::例2

Find the value of the discriminant and state the types of solutions.
::找出争议方的价值,说明解决办法的类型。

$\begin{aligned} 3 x^{2} - 5 x - 12 = 0 \end{aligned}$
$\begin{align*}3x^2-5x-12=0\end{align*}$
::3x2 - 5x - 12=0

Solution: Use the discriminant. a = 3, b = -5, and c = -12
::解答:使用辨别词。 a = 3, b = 5, c = - 12

$\begin{aligned} \sqrt{(- 5)^{2} - 4 (3) (- 12)} = \sqrt{25 + 144} = \sqrt{169} = 13 \end{aligned}$
$\begin{align*} \sqrt{(\text{-}5)^2 - 4(3)(\text{-}12)}=\sqrt{25+144}=\sqrt{169}=13\end{align*}$

This equation has two real solutions.
::这个方程式有两个真正的解决办法。

Example 3
::例3

Use the discriminant to determine the type of solutions $\begin{align*}25x^2-80x+64=0\end{align*}$ has.
::使用 Dispriminant 来确定 25x2- 80x+64=0 的解决方案类型。

$\begin{aligned} b^{2} - 4 a c & = (- 80)^{2} - 4 (25) (64) \\ = 6, 400 - 6, 400 \\ = 0 \end{aligned}$
$\begin{align*}b^2-4ac &=(\text{-}80)^2-4(25)(64)\\ &=6,\!400-6,\!400\\ &=0\end{align*}$
::b2-4ac=(-80)2-4(25)(64)=6,400-6,400=0

This equation has one real solution.
::这个方程式有一个真正的解决方案

Solving Quadratic Equations With Complex Solutions
::以复杂解决方案解决赤道赤道

Example 4
::例4

Solve $\begin{align*}3x^2+27=0\end{align*}$ .
::解决 3x2+27=0。

Solution: First, factor out the GCF.
::解决办法:首先,考虑一下全球合作框架。

$\begin{aligned} 3 (x^{2} + 9) = 0 \end{aligned}$
$\begin{align*}3(x^2+9)=0\end{align*}$
::3(x2+9)=0

Now try to factor $\begin{align*}x^2 + 9\end{align*}$ . Rewrite the quadratic as $\begin{align*}x^2 + 0x + 9\end{align*}$ to help. There are no factors of 9 that add up to 0. Therefore, this is not a factorable quadratic. Let's solve it using square roots.
::现在尝试将二次曲线重写为 x2+0x+9 以起到帮助作用。没有9 的系数加起来等于 0。因此, 这不是一个可乘因数的二次曲线。让我们用正方根来解决它。

$\begin{aligned} 3 (x^{2} + 9) & = 0 Divide both sides by 3. \\ x^{2} + 9 & = 0 \\ x^{2} & = - 9 \\ x & = \pm \sqrt{- 9} = \pm 3 i \end{aligned}$
$\begin{align*}3(x^2+9 )&= 0 \qquad \qquad \color{gray}{\text{Divide both sides by 3.}}\\ x^2+9 &= 0\\ x^2 &= \text{-}9\\ x &= \pm \sqrt{\text{-}9}= \pm 3i\end{align*}$
::3(x2+9) =0 双侧除以 3.x2+9=0x2=9x=9x- 93i

Example 5
::例5

Solve $\begin{align*}(x-8)^2=\text{-}25\end{align*}$ .
::解决(x-8)2=25。

Solution: Solve using square roots.
::解决办法:用平方根解决。

$\begin{aligned} (x - 8)^{2} & = - 25 \\ x - 8 & = \pm 5 i \\ x & = 8 \pm 5 i \end{aligned}$
$\begin{align*}(x-8)^2 &= \text{-}25\\ x-8 &= \pm 5i\\ x &= 8 \pm 5i\end{align*}$
:x-8)2=-25x-8=85ix=85i

Example 6
::例6

Solve $\begin{align*}\text{-}\frac{1}{2}(3x+8)^2-16=2\end{align*}$ .
::解决 - 12( 3x+8) 2 - 16=2。

Solution: Since there is a squared term, we can use the square root method.
::解决方案:既然有一个正方形的术语, 我们可以使用平方根法。

$\begin{aligned} - \frac{1}{2} (3 x + 8)^{2} - 16 & = 2 \\ - \frac{1}{2} (3 x + 8)^{2} & = 18 \\ (3 x + 8)^{2} & = - 36 \\ 3 x + 8 & = \pm 6 i \\ 3 x & = - 8 \pm 6 i \\ x & = - \frac{8}{3} \pm 2 i \end{aligned}$
$\begin{align*}\text{-}\frac{1}{2}(3x+8)^2-16 &=2\\ \text{-}\frac{1}{2}(3x+8)^2 &=18\\ (3x+8)^2 &=\text{-}36\\ 3x+8 &=\pm6i\\ 3x &=\text{-}8\pm6i\\ x &=\text{-}\frac{8}{3}\pm2i\end{align*}$
::-12(3x+8)2--16=2-12(3x+8)2=18(3x+8)2=-363x+8=-363x+8*6i3x=-8_6x=-8_6ix=-83_2i

by Tyler Wallace demonstrates how to complete the square in examples with irrational or complex solutions.
::泰勒·华莱士(Tyler Wallace)以非理性或复杂的解决方案为例,展示如何完成广场。

Example 7
::例7

Solve the equation $\begin{align*}4x^2-5x+17=0\end{align*}$ to prove that it does have two complex solutions.
::解决公式 4x2 - 5x+17=0 以证明它确实有两个复杂的解决方案。

Solution: Use the Quadratic Formula.
::解决方案:使用二次曲线公式。

$\begin{aligned} x = \frac{5 \pm \sqrt{25 - 272}}{8} = \frac{5 \pm \sqrt{- 247}}{8} = \frac{5}{8} \pm \frac{\sqrt{247}}{8} i \end{aligned}$
$\begin{align*}x = \frac{5 \pm \sqrt{25-272}}{8}=\frac{5 \pm \sqrt{\text{-}247}}{8}= \frac{5}{8} \pm \frac{\sqrt{247}}{8}i\end{align*}$
::x=5=25-2728=5=2478=58=2478i

Example 8
::例8

Bryson owns a business that manufactures and sells tires. The revenue from selling tires in the month of July is given by the function $\begin{align*}R(x)=x(200-0.4x)\end{align*}$ , where $\begin{align*}x\end{align*}$ is the number of tires sold. Can Bryson's business generate a revenue of $30,000 in the month of July?
::Bryson拥有一个制造和销售轮胎的企业,7月份销售轮胎的收入由R(x)=x(200-0.4x)函数提供,而售出的轮胎数量是x。 Bryson的生意在7月份能否产生3万美元的收入?

Solution: We can replace $\begin{align*}R(x)\end{align*}$ by 30,000 and determine the discriminant . If there is a real solution, then it is possible. If the solutions are complex, then it will be impossible.
::解决方案 : 我们可以用 30,000 来取代 R( x) , 并且确定对立点。如果有真正的解决方案, 那么它就有可能。如果解决方案复杂, 那么它就是不可能的。

$\begin{aligned} 30, 000 & = x (200 - 0.4 x) \\ 30, 000 & = 200 x - 0.4 x^{2} \\ 0 = 200 x & - 0.4 x^{2} - 30, 000 \\ 0 = - 0.4 x^{2} & + 200 x - 30, 000 \\ b^{2} - 4 a c & = 200^{2} - 4 (- 0.4) (- 30, 000) \\ = 40, 000 - 48, 000 \\ = - 8, 000 \end{aligned}$
$\begin{align*}30,\!000&=x(200-0.4x) \\ 30,\!000&=200x-0.4x^2 \\ 0=200x&-0.4x^2-30,\!000 \\ 0=-0.4x^2&+200x-30,\!000\\ \\ b^2-4ac&=200^2-4(\text{-}0.4)(\text{-}30,\!000)\\ &=40,\!000-48,\!000\\ &=\text{-}8,\!000\end{align*}$
::30,000=x(200-0.4x)30,000=200x-0.4x20=200x-0.4x20=200x-0.4x2-3000=0.4x2+200x-300b2-4ac=2002-4(-0.4)(30,000)=40,000-48,000=-8,000

Since the discriminant is negative, there will be no real solutions to this quadratic equation. Bryson's business cannot generate $30,000 in revenue in July.
::因为争议是负面的,所以这个二次等式不会有真正的解决方案。布赖森的生意在7月无法产生30,000美元的收入。

Summary
::摘要

The discriminant, $\begin{align*}b^2-4ac \end{align*}$ , indicates the types of solutions of a quadratic equation.
::争议 b2 -4ac 表示二次方程的解决方案类型。

::争议 b2 -4ac 表示二次方程的解决方案类型。

If the discriminant equals 0, then the equation has one real solution, a double root.
::如果对立方程式等于0, 那么方程式就有一个真正的解决方案, 一个双根。

::如果对立方程式等于0, 那么方程式就有一个真正的解决方案, 一个双根。

If the discriminant is less than 0, then the equation has two complex solutions.
::如果对立方程式小于0,那么方程式有两个复杂的解决方案。

::如果对立方程式小于0,那么方程式有两个复杂的解决方案。

If the discriminant is greater than 0, then the equation has two real solutions.
::如果对立方程式大于0,那么方程式有两个真正的解决方案。

::如果对立方程式大于0,那么方程式有两个真正的解决方案。

Review
::回顾

Determine the number and type of solutions each equation has.
::确定每个方程式的解决方案的数量和类型。

1. $\begin{align*}x^2-12x+36=0\end{align*}$
::1. x2 - 12x+36=0

2. $\begin{align*}5x^2-9=0\end{align*}$
::2. 5x2-9=0

3. $\begin{align*}2x^2+6x+15=0\end{align*}$
::3. 2x2+6x+15=0

4. $\begin{align*}\text{-}6x^2+8x+21=0\end{align*}$
::4. -6x2+8x+21=0

Solve.
::解决。

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

6. $\begin{align*}x^2-17x-60=0\end{align*}$
::6. x2-17x-60=0

7. $\begin{align*}(x+1)^2=\text{-}121\end{align*}$
::7. (x+1)2=-121

8. $\begin{align*}6x^2-20=0\end{align*}$
::8. 6x2-20=0

9. $\begin{align*}5x^2+16=\text{-}29\end{align*}$
::9. 5x2+16=-29

10. $\begin{align*}14-4x^2=38\end{align*}$
::10. 14-4x2=38

11. $\begin{align*}2x^2+5x+11=0\end{align*}$
::11. 2x2+5x+11=0

12. $\begin{align*}\text{-}3(x+6)^2+1=37\end{align*}$
::-12.3(x+6)2+1=37

13. $\begin{align*}(2x-1)^2+5=\text{-}23\end{align*}$
::13. (2x-1)2+5=23

14. $\begin{align*}\text{-}(6x+5)^2=72\end{align*}$
::14.(6x+5)2=72

15. $\begin{align*}7(4x-3)^2-15=\text{-}68\end{align*}$
::15. 7(4x-3-3)2-15=-68

Explore More
::探索更多

1a. If a quadratic equation has $\begin{align*}4 - i\end{align*}$ as a solution, what must the other solution be?
::1.a. 如果四方方程式有4-i作为解决办法,其他解决办法又是什么?

1b. If a quadratic equation has $\begin{align*}6 + 2i\end{align*}$ as a solution, what must the other solution be?
::1.b. 如果四方方程式有6+2i作为解决办法,其他解决办法又是什么?

2. Recall that the factor of a quadratic equation has the form $\begin{align*}(x\pm m),\end{align*}$ where $\begin{align*}m\end{align*}$ is any number.
::2. 回顾二次方程的因数为表(xm),m为任意数字。

2a. Find a quadratic equation that has the solution $\begin{align*}3 + 2i\end{align*}$ .
::2.a. 找到一个具有3+2i解决方案的二次方程。

2b. Find a quadratic equation that has the solution $\begin{align*}1-i\end{align*}$ .
::2.b. 找到一个具有第1-i项解决办法的二次方程。

3. Determine the values for $\begin{align*}c\end{align*}$ that make the equation have either two real solutions, one real solution, or two complex solutions.
::3. 确定使等式具有两种实际解决办法、一种实际解决办法或两种复杂解决办法的c值。

a. $\begin{align*}x^2+2x+c=0\end{align*}$
::a. x2+2x+c=0

b. $\begin{align*}x^2-6x+c=0\end{align*}$
::b. b. x2-6x+c=0

4. Emma and Brandon own a factory that produces bike helmets. Their accountant says that their profit per year is given by the function $\begin{align*}P=0.003x^2+12x+27,\!760\end{align*}$ , where x represents the number of helmets produced. Their goal is to make a profit of $40,000 this year. Is this possible?
::4. Emma和Brandon拥有一家生产自行车头盔的工厂,他们的会计说,他们每年的利润由P=0.003x2+12x+27,760函数提供,其中x表示生产的头盔数量,他们的目标是今年赚取4万美元利润,这是可能的吗?

by CK-12 demonstrates an application of the discriminant .
::CK-12展示了对持不同政见者的适用。

5. Marty is outside his apartment building. He needs to give his roommate Yolanda her cell phone, but he doesn't have time to run upstairs to the 3rd floor to give it to her. He throws it straight up with a vertical velocity of 55 feet/second. Will the phone reach her if she is 36 feet up? (Hint: The equation for the height is $\begin{align*}y=\text{-}32t^2+55t+4\end{align*}$ .)
::5. Marty在公寓大楼外面,他需要给他的室友Yolanda她的手机,但他没有时间跑上楼三楼给她手机,他直直往上扔,垂直速度为55英尺/秒,如果她高36英尺,电话会接上她吗? (提示:高度的方程式是y=-32t2+55t+4。 )

6. Error Analysis. Find the mistake in student A's work and correct it. Find the solution set of the given quadratic equation.
::6. 错误分析. 找出A学生工作中的错误并加以纠正. 找出给定的四方形方程式的解决方案集。

Student A's work:
::学生A的工作:

$\begin{aligned} \frac{x^{2}}{2} & = 6 x - 19 \\ Step 1 x & = \frac{- (- 12) \pm \sqrt{(- 12)^{2} - 4 (1) (38)}}{2 (1)} \\ Step 2 x & = \frac{12 \pm \sqrt{144 + 152}}{2} \\ Step 3 x & = \frac{12 \pm \sqrt{296}}{2} \\ Step 4 x & = 6 \pm \sqrt{74} \end{aligned}$
$\begin{align*}\frac{x^2}{2} &= 6x - 19 \\ \text{Step 1} \qquad x &= \frac{\text{-}(\text{-}12) \pm \sqrt{(\text{-}12)^2 - 4(1)(38)}}{2(1)} \\ \\ \text{Step 2} \qquad x &= \frac{12 \pm \sqrt{144 + 152}}{2} \\ \\ \text{Step 3} \qquad x &= \frac{12 \pm \sqrt{296}}{2} \\ \\ \text{Step 4} \qquad x &= 6 \pm \sqrt{74}\end{align*}$
::x22=6x-19Step 1x=-(-12)-(-12)-(-12)-(-12)-2-4(1)(38)2(1) Step 2x=12×144+1522 Step 3x=12×2962 Step 4x=6×74

7. Is it possible to have one real solution and one complex solution to a quadratic equation with real number coefficients? Explain.
::7. 能否对具有实际数字系数的二次方程式有一个真正的解决办法和一个复杂的解决办法?

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

Please see the Appendix.
::请参看附录。

PLIX
::PLIX

Try these interactives that reinforce the concepts explored in this section:
::尝试这些强化本节所探讨概念的交互作用 :

章节大纲

The Discriminant ::诽谤者,

Investigation: Solving Equations with Different Types of Solutions ::调查:用不同类型解决办法解决等式问题

Determining the Number and Types of Solutions From the Discriminant ::确定来自争议方的解决办法的数目和类型

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Solving Quadratic Equations With Complex Solutions ::以复杂解决方案解决赤道赤道

Example 4 ::例4

Example 5 ::例5

Example 6 ::例6

Example 7 ::例7

Example 8 ::例8

Summary ::摘要

Review ::回顾

Explore More ::探索更多

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

PLIX ::PLIX

The Discriminant
::诽谤者,

Investigation: Solving Equations with Different Types of Solutions
::调查:用不同类型解决办法解决等式问题

Determining the Number and Types of Solutions From the Discriminant
::确定来自争议方的解决办法的数目和类型

Example 1
::例1

Example 2
::例2

Example 3
::例3

Solving Quadratic Equations With Complex Solutions
::以复杂解决方案解决赤道赤道

Example 4
::例4

Example 5
::例5

Example 6
::例6

Example 7
::例7

Example 8
::例8

Summary
::摘要

Review
::回顾

Explore More
::探索更多

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

PLIX
::PLIX