课程： 5.14. 使用公诉人

章节大纲

选择节常规

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选择节使用争议

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使用争议
The profit on your school fundraiser is represented by the quadratic expression $\begin{align*}-5p^2 + 400p - 8000\end{align*}$ , where p is your price point. How many real break-even points will you have?
::您的学校募捐活动的利润由 5p2+400p- 8000 的二次表达式表示, p 是您的价格点。您有多少真正的平衡点 ?

Discriminant
::持不同意见者

Remember, t he Quadratic Formula is $\begin{align*}x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\end{align*}$ . 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.
::记住, 二次曲线公式是 xbb2- 4ac2a 。激进表达式 b2-4ac 下的表达式被称为 disriminant 。您可以使用 dispriminant 来确定方程式的解决方案数量和类型。

Solving Equations with Different Types of Solutions
::解决不同类型解决办法的等式

Step 1: Solve $\begin{align*}x^2-8x-20=0\end{align*}$ using . What is the value of the discriminant?
::步骤 1: 使用 . 解决 x2 - 8x - 20=0 。对话者的价值是多少 ?

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

Step 2: Solve $\begin{align*}x^2-8x+6=0\end{align*}$ using the Quadratic Formula. What is the value of the discriminant?
::步数 2: 使用二次曲线公式解决 x2 - 8x+6=0。共振值是多少 ?

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

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

$\begin{aligned} x & = \frac{8 \pm \sqrt{- 16}}{2} \\ = \frac{8 \pm 4 i}{2} \to 4 \pm 2 i \end{aligned}$
$\begin{align*}x &=\frac{8 \pm \sqrt{{\color{red}-16}}}{2}\\ &=\frac{8 \pm 4i}{2} \rightarrow 4 \pm 2i\end{align*}$
::x8=___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Step 4: Look at the values of the discriminants from Steps 1-3. How do they differ? How does that affect the final answer?
::第4步:看看第1至第3步的抗议者的价值观。他们有什么不同?这对最终答案有什么影响?

From the steps above, we can conclude:
::从上述步骤中,我们可以得出以下结论:

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。

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; 双根。

If $\begin{align*}b^2-4ac < 0\end{align*}$ , then the equation has two imaginary solutions.
::如果 b2 - 4ac < 0, 那么方程式有两个假想的解决方案。

Determine the type of solutions for the equation.
::确定方程的解决方案类型。

$\begin{aligned} 4 x^{2} - 5 x + 17 = 0 \end{aligned}$
$\begin{align*}4x^2-5x+17=0\end{align*}$
::4x2-5x+17=0

Find the discriminant.
::找到异见者

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

At this point, we know the answer is going to be negative, so there is no need to continue (unless we were solving the problem). This equation has two imaginary solutions.
::此时此刻,我们知道答案将是否定的,所以没有必要继续(除非我们正在解决问题 ) 。这个等式有两个想象中的解决方案。

Use the Quadratic Formula for the previous problem.
::对上一个问题使用二次曲线公式。

$\begin{aligned} x = \frac{5 \pm \sqrt{25 - 272}}{8} = \frac{5 \pm \sqrt{- 247}}{8} = \frac{5}{8} \pm \frac{\sqrt{247}}{8} i \end{aligned}$
$\begin{align*}x = \frac{5 \pm \sqrt{25-272}}{8}=\frac{5 \pm \sqrt{-247}}{8}= \frac{5}{8} \pm \frac{\sqrt{247}}{8}i\end{align*}$
::x=525-2728=52478=582478i

Determine the type of solutions for the equation.
::确定方程的解决方案类型。

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

Use the discriminant. a = 3, b = -5, and c = -12
::a=3,b=5,c=12。

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

This quadratic has two real solutions.
::这个二次曲线有两个真正的解决方案。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the number of break-even points will you have.
::早些时候,有人要求你找出你能得到的平衡点数。

Set the expression $\begin{align*}-5p^2 + 400p - 8000\end{align*}$ equal to zero and then find the discriminant.
::设置表达式 - 5p2+400p- 8000 等于零, 然后找到 driminant 。

$\begin{align*}-5p^2 + 400p - 8000 = 0\end{align*}$
::-5p2+400p-800=0

$\begin{aligned} b^{2} - 4 a c & = (400)^{2} - 4 (- 5) (- 8000) \\ = 160000 - 160000 = 0 \end{aligned}$
$\begin{align*}b^2-4ac &=(400)^2-4(-5)(-8000)\\ &=160000-160000 = 0\end{align*}$
::b2-4ac=(4000)-2-4(5)(-800)=160000-160000=0)

At this point, we know the answer is zero, so the equation has one real solution. Therefore , there is one real break-even point.
::此时此刻,我们知道答案是零,所以方程式有一个真正的解决方案。因此,有一个真正的平衡点。

Example 2
::例2

Use the discriminant to determine the type of solutions $\begin{align*}-3x^2-8x+16=0\end{align*}$ has.
::使用辨识器确定 - 3x2-8x+16=0 的解决方案类型。

$\begin{aligned} b^{2} - 4 a c & = (- 8)^{2} - 4 (- 3) (16) \\ = 64 + 192 \\ = 256 \end{aligned}$
$\begin{align*}b^2-4ac &=(-8)^2-4(-3)(16)\\ &=64+192\\ &=256\end{align*}$
::b-2-4ac=(-8)2-4(-3)(16)=64+192=256

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。

$\begin{aligned} b^{2} - 4 a c & = (- 80)^{2} - 4 (25) (64) \\ = 6400 - 6400 \\ = 0 \end{aligned}$
$\begin{align*}b^2-4ac &=(-80)^2-4(25)(64)\\ &=6400-6400\\ &=0\end{align*}$
::b2-4ac=(-802)-4(25)(64)=6400-6400=0

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

Example 4
::例4

Solve the equation from Example 2.
::从例2中解决方程式。

$\begin{align*}x = \frac{8 \pm \sqrt{256}}{-6}=\frac{8 \pm 16}{-6}=-4, \frac{4}{3}\end{align*}$
::x=8256-6=816-64,43

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。

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

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

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

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

$\begin{align*}x^2+15x+26=0\end{align*}$
::x2+15x+26=0

$\begin{align*}4x^2+x+1=0\end{align*}$
::4x2+x+1=0

Solve the following equations using the Quadratic Formula.
::使用“二次曲线公式”解决以下方程式。

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

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

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

Challenge Determine the values for $\begin{align*}c\end{align*}$ that make the equation have a) two real solutions, b) one real solution, and c) two imaginary solutions.
::确定使等式具有以下两种实际解决办法的c值的挑战,a) 两种实际解决办法,b) 一种实际解决办法,c) 两种假想解决办法。

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

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

$\begin{align*}x^2+12x+c=0\end{align*}$
::x2+12x+c=0

What is the discriminant of $\begin{align*}x^2+2kx+4=0\end{align*}$ ? Write your answer in terms of $\begin{align*}k\end{align*}$ .
::x2+2kx+4=0的辩驳词是什么? 以 k 写您的回答。

For what values of $\begin{align*}k\end{align*}$ will the equation have two real solutions?
::K 的值会是什么? 等式会有两种真正的解决方案吗?

For what values of $\begin{align*}k\end{align*}$ will the equation have one real solution?
::方程式的K值能有一个真正的解决方案吗?

For what values of $\begin{align*}k\end{align*}$ will the equation have two imaginary solutions?
::K 的值为何方程式会有两种假想的解决方案?

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

使用争议

Discriminant ::持不同意见者

Solving Equations with Different Types of Solutions ::解决不同类型解决办法的等式

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Discriminant
::持不同意见者

Solving Equations with Different Types of Solutions
::解决不同类型解决办法的等式

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)