Course: 10.10 利用争议解决

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利用争议解决
Solutions Using the Discriminant
::利用争议解决

In the quadratic formula , $\begin{aligned} x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}$ , the expression inside the square root is called the discriminant . The discriminant can be used to analyze the types of solutions to a quadratic equation without actually solving the equation . Here’s how:
::在四方形 xbb2 - 4ac2a 中,平方根内的表达式被称为 disriminant。共振可以用来分析四方形的解决方案类型, 而不实际解析方程式。以下是 :

If $\begin{aligned} b^{2} - 4 a c > 0 \end{aligned}$ , the equation has two separate real solutions.
::如果 b2 - 4ac>0, 方程式有两个不同的真实解决方案。

If $\begin{aligned} b^{2} - 4 a c < 0 \end{aligned}$ , the equation has only non-real solutions.
::如果 b2 - 4ac < 0, 方程式只有非实际的解决方案。

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

Find the Discriminant of a Quadratic Equation
::查找二次赤道的偏差

To find the discriminant of a quadratic equation we calculate $\begin{aligned} D = b^{2} - 4 a c \end{aligned}$ .
::为了找到二次方程的分歧我们计算了D=b2 -4ac

Find the discriminant of each quadratic equation. Then tell how many solutions there will be to the quadratic equation without solving.
::找到每个二次方程的对立方程。然后告诉您在不解决的情况下, 二次方程会有多少个解决方案。

a) $\begin{aligned} x^{2} - 5 x + 3 = 0 \end{aligned}$
::a) x2 - 5x+3=0

Plug $\begin{aligned} a = 1, b = - 5 \end{aligned}$ and $\begin{aligned} c = 3 \end{aligned}$ into the discriminant formula : $\begin{aligned} D = (- 5)^{2} - 4 (1) (3) = 13 \end{aligned}$ $\begin{aligned} D > 0 \end{aligned}$ , so there are two real solutions.
::将 a=1, b5 和 c=3 插入对立公式: D=(-5)2-4(1)(3)=13D>0, 所以有两种真正的解决方案。

b) $\begin{aligned} 4 x^{2} - 4 x + 1 = 0 \end{aligned}$
:b) 4x2-4x+1=0

Plug $\begin{aligned} a = 4, b = - 4 \end{aligned}$ and $\begin{aligned} c = 1 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (- 4)^{2} - 4 (4) (1) = 0 \end{aligned}$ $\begin{aligned} D = 0 \end{aligned}$ , so there is one real solution.
::将 a = 4, b4 和 c=1 插入对立公式: D=(-4)2-4(4)(1)=0D=0,所以有一个真正的解决方案。

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

Rewrite the equation in standard form : $\begin{aligned} - 2 x^{2} + x - 4 = 0 \end{aligned}$
::以标准格式重写方程式:- 2x2+x- 4=0

Plug $\begin{aligned} a = - 2, b = 1 \end{aligned}$ and $\begin{aligned} c = - 4 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (1)^{2} - 4 (- 2) (- 4) = - 31 \end{aligned}$ $\begin{aligned} D < 0 \end{aligned}$ , so there are no real solutions.
::将a2、b=1和c4插入对立公式:D=(1)2-4(-2)(-4)(4)31D <0,所以没有真正的解决办法。

Interpret the Discriminant of a Quadratic Equation
::解释二次等同的辩驳词

The sign of the discriminant tells us the nature of the solutions (or roots) of a quadratic equation. We can obtain two distinct real solutions if $\begin{aligned} D > 0 \end{aligned}$ , two non-real solutions if $\begin{aligned} D < 0 \end{aligned}$ or one solution (called a double root) if $\begin{aligned} D = 0 \end{aligned}$ . Recall that the number of solutions of a quadratic equation tells us how many times its graph crosses the $\begin{aligned} x - \end{aligned}$ axis. If $\begin{aligned} D > 0 \end{aligned}$ , the graph crosses the $\begin{aligned} x - \end{aligned}$ axis in two places; if $\begin{aligned} D = 0 \end{aligned}$ it crosses in one place; if $\begin{aligned} D < 0 \end{aligned}$ it doesn’t cross at all:
::差异符号告诉我们四方形的解决方案( 或根) 的性质。如果 D > 0, 如果 D < 0 或 D=0 一种解决方案( 称为双根) , 我们可以获得两个截然不同的真正解决方案。回顾四方形的解决方案数量告诉我们其图形横跨 x - 轴的几倍。如果 D > 0 , 图形横跨两个地方的 x - 轴; 如果 D= 0 跨一个地方; 如果 D < 0 根本不交叉 :

Determining the Nature of Solutions
::确定解决办法的性质

1. Determine the nature of the solutions of each quadratic equation.
::1. 确定每个二次方程解决办法的性质。

Use the value of the discriminant to determine the nature of the solutions to the quadratic equation.
::使用分歧方程式的价值来确定四方形解决方案的性质。

a) $\begin{aligned} 4 x^{2} - 1 = 0 \end{aligned}$
:a) 4x2-1=0

Plug $\begin{aligned} a = 4, b = 0 \end{aligned}$ and $\begin{aligned} c = - 1 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (0)^{2} - 4 (4) (- 1) = 16 \end{aligned}$
::将 a=4, b=0 和 c1 插入对称公式中: D=( 0)2-4(4)(-1)=16

The discriminant is positive, so the equation has two distinct real solutions.
::争议是积极的,所以等式有两个截然不同的实际解决办法。

The solutions to the equation are: $\begin{aligned} \frac{0 \pm \sqrt{16}}{8} = \pm \frac{4}{8} = \pm \frac{1}{2} \end{aligned}$
::等式的解决方案是: 01684812

b) $\begin{aligned} 10 x^{2} - 3 x = - 4 \end{aligned}$
:b) 10x2-3x%4

Re-write the equation in standard form: $\begin{aligned} 10 x^{2} - 3 x + 4 = 0 \end{aligned}$
::以标准格式重写方程: 10x2- 3x+4=0

Plug $\begin{aligned} a = 10, b = - 3 \end{aligned}$ and $\begin{aligned} c = 4 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (- 3)^{2} - 4 (10) (4) = - 151 \end{aligned}$
::将 a = 10, b = 3 和 c= 4 插入对立公式: D = (-3)2 - 4(10) (4) = 151

The discriminant is negative, so the equation has two non-real solutions.
::争议是负面的,所以等式有两个非现实的解决办法。

c) $\begin{aligned} x^{2} - 10 x + 25 = 0 \end{aligned}$
:c) x2-10x+25=0

Plug $\begin{aligned} a = 1, b = - 10 \end{aligned}$ and $\begin{aligned} c = 25 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (- 10)^{2} - 4 (1) (25) = 0 \end{aligned}$
::将 a= 1, b 10 和 c= 25 插入对立公式中: D= (- 10)2-4(1)( 25)= 0

The discriminant is 0, so the equation has a double root.
::争议是0, 所以方程式有一个双根。

The solution to the equation is: $\begin{aligned} \frac{10 \pm \sqrt{0}}{2} = \frac{10}{2} = 5 \end{aligned}$
::方程的解决方案是: 1002=102=5

If the discriminant is a perfect square , then the solutions to the equation are not only real, but also rational. If the discriminant is positive but not a perfect square, then the solutions to the equation are real but irrational.
::如果对立方是完美的正方形,那么方程式的解决方案不仅是真实的,而且是理性的。如果对立方形是正面的,但不是完美的正方形,那么对等方形的解决方案是真实的,但非理性的。

2. Determine the nature of the solutions to each quadratic equation.
::2. 确定每个二次方程式解决办法的性质。

Use the discriminant to determine the nature of the solutions.
::使用歧见来确定解决办法的性质。

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

Plug $\begin{aligned} a = 2, b = 1 \end{aligned}$ and $\begin{aligned} c = - 3 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (1)^{2} - 4 (2) (- 3) = 25 \end{aligned}$
::将 a=2, b=1 和 c=3 插入相撞公式: D=(1)2-4(2)-(3)=25

The discriminant is a positive perfect square, so the solutions are two real rational numbers.
::争议是一个积极的完全的正方形,所以解决办法是两个真正的合理数字。

The solutions to the equation are: $\begin{aligned} \frac{- 1 \pm \sqrt{25}}{4} = \frac{- 1 \pm 5}{4} \end{aligned}$ , so $\begin{aligned} x = 1 \end{aligned}$ and $\begin{aligned} x = - \frac{3}{2} \end{aligned}$ .
::等式的解决方案是: - 1254154, 所以 x=1 和 x32。

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

Plug $\begin{aligned} a = 5, b = - 1 \end{aligned}$ and $\begin{aligned} c = - 1 \end{aligned}$ into the discriminant formula: $\begin{aligned} D = (- 1)^{2} - 4 (5) (- 1) = 21 \end{aligned}$
::将 a = 5, b 1 和 c 1 插入对立公式: D = (-1) 2-4(5) (-1) = 21

The discriminant is positive but not a perfect square, so the solutions are two real irrational numbers.
::争议是积极的,但不是完美的平方,所以解决办法是两个真正的非理性数字。

The solutions to the equation are: $\begin{aligned} \frac{1 \pm \sqrt{21}}{10} \end{aligned}$ , so $\begin{aligned} x \approx 0.56 \end{aligned}$ and $\begin{aligned} x \approx - 0.36 \end{aligned}$ .
::方程式的解决方案是: 1211, 所以 x0.56 和 x0. 36。

Solve Real-World Problems Using Quadratic Functions and Interpreting the Discriminant
::利用四分法函数和解释争议解释解决现实世界问题

You’ve seen that calculating the discriminant shows what types of solutions a quadratic equation possesses. Knowing the types of solutions is very useful in applied problems. Consider the following situation.
::您已经看到,计算分歧的计算表明四方形拥有什么样的解决方案。了解哪类解决方案对于应用问题非常有用。考虑以下情形。

Real-World Application: Football
::真实世界应用:足球

Marcus kicks a football in order to score a field goal. The height of the ball is given by the equation $\begin{aligned} y = - \frac{32}{6400} x^{2} + x \end{aligned}$ . If the goalpost is 10 feet high, can Marcus kick the ball high enough to go over the goalpost? What is the farthest distance that Marcus can kick the ball from and still make it over the goalpost?
::Marcus踢足球是为了得分球。球的高度是由 y 326400x2+x 等式给出的。如果目标柱高10 英尺, Marcus 能够踢得高到可以越过目标柱吗? Marcus可以踢球到最远的距离是多少?

Define: Let $\begin{aligned} y = \end{aligned}$ height of the ball in feet.
::定义:让Y=球脚高度。

Let $\begin{aligned} x = \end{aligned}$ distance from the ball to the goalpost.
::让我们 x= 从球到目标柱的距离。

Translate: We want to know if it is possible for the height of the ball to equal 10 feet at some real distance from the goalpost.
::翻译:我们想知道球的高度是否可能等于10英尺距离球门的距离。

Solve:
::解决 :

$\begin{aligned} Write the equation in standard form: & - \frac{32}{6400} x^{2} + x - 10 = 0 \\ Simplify: & - 0.005 x^{2} + x - 10 = 0 \\ Find the discriminant: & D = (1)^{2} - 4 (- 0.005) (- 10) = 0.8 \end{aligned}$

::以标准格式写入方程式 : - 326400x2+x- 10=0 简化 : - 0.005x2+x- 10=0 查找辨识器 D=(1)2-4(- 0.005)(- 10)=0. 8

Since the discriminant is positive, we know that it is possible for the ball to go over the goal post, if Marcus kicks it from an acceptable distance $\begin{aligned} x \end{aligned}$ from the goalpost.
::由于对立是正面的, 我们知道球有可能越过目标柱, 如果马库斯踢它从一个可接受的距离x 与目标柱。

To find the value of $\begin{aligned} x \end{aligned}$ that will work, we need to use the quadratic formula:
::要找到x的值,这将会奏效, 我们需要使用二次公式 :

$\begin{aligned} x = \frac{- 1 \pm \sqrt{0.8}}{- 0.01} = 189.4 f e e t or 10.56 f e e t \end{aligned}$

::x10.8-0.01=189.4英尺或10.56英尺

What does this answer mean? It means that if Marcus is exactly 189.4 feet or exactly 10.56 feet from the goalposts, the ball will just barely go over them. Are these the only distances that will work? No; those are just the distances at which the ball will be exactly 10 feet high, but between those two distances, the ball will go even higher than that. (It travels in a downward-opening parabola from the place where it is kicked to the spot where it hits the ground.) This means that Marcus will make the goal if he is anywhere between 10.56 and 189.4 feet from the goalposts.
::这个答案意味着什么?如果马库斯的距离是189.4英尺或10.56英尺,那么球就几乎不会超过它们。这些是唯一能起作用的距离吗?不,这些只是球在10.56至189.4英尺之间的距离,但是在这两英尺之间的距离,球会比这还要高。 (它从被踢到地面的地方,用一个向下开的抛物线运行。 )这意味着,如果马库斯在距目标柱10.56至189.4英尺之间,就会达到目标。

Example
::示例示例示例示例

Example 1
::例1

Emma and Bradon own a factory that produces bike helmets. Their accountant says that their profit per year is given by the function $\begin{aligned} P = - 0.003 x^{2} + 12 x + 27760 \end{aligned}$ , where $\begin{aligned} x \end{aligned}$ is the number of helmets produced. Their goal is to make a profit of $40,000 this year. Is this possible?
::Emma和Bradon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来自P0.003x2+12x+27760, 其中x生产了多少头盔。他们的目标是今年赚取40,000美元的利润。这有可能吗?

We want to know if it is possible for the profit to equal $40,000.
::我们想知道,能否使利润达到4万美元。

$\begin{aligned} 40000 = - 0.003 x^{2} + 12 x + 27760 \end{aligned}$

::400000 0.003x2+12x+27760

Write the equation in standard form: $\begin{aligned} - 0.003 x^{2} + 12 x - 12240 = 0 \end{aligned}$
::以标准格式写入方程式 : - 0.003x2+12x- 12240=0

Find the discriminant: $\begin{aligned} D = (12)^{2} - 4 (- 0.003) (- 12240) = - 2.88 \end{aligned}$
::查找对立物:D=(12)2-4(-0.003)(-12240)_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Since the discriminant is negative, we know that it is not possible for Emma and Bradon to make a profit of $40,000 this year no matter how many helmets they make.
::由于争议是负面的,我们知道,艾玛和布拉登今年不可能赚取4万美元利润,不管他们制造多少头盔。

Review
::回顾

Find the discriminant of each quadratic equation.
::找到每个二次方程的分歧点

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

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

$\begin{aligned} 4 x^{2} - 12 x + 9 = 0 \end{aligned}$
::4x2 - 12x+9=0

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

$\begin{aligned} x^{2} - 16 x = 32 \end{aligned}$
::x2 - 16x=32

$\begin{aligned} - 5 x^{2} + 5 x - 6 = 0 \end{aligned}$
::-5x2+5x-6=0

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

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

Determine the nature of the solutions of each quadratic equation.
::确定每个二次方程的解决方案的性质。

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

$\begin{aligned} 5 x^{2} = 6 x \end{aligned}$
::5x2=6x

$\begin{aligned} 41 x^{2} - 31 x - 52 = 0 \end{aligned}$
::41x2-31x-52=0

$\begin{aligned} x^{2} - 8 x + 16 = 0 \end{aligned}$
::x2-8x+16=0

$\begin{aligned} - x^{2} + 3 x - 10 = 0 \end{aligned}$
::-x2+3x-10=0

$\begin{aligned} x^{2} - 64 = 0 \end{aligned}$
::x2- 64=0

$\begin{aligned} 3 x^{2} = 7 \end{aligned}$
::3x2=7

$\begin{aligned} x^{2} + 30 + 225 = 0 \end{aligned}$
::x2+30+225=0

Without solving the equation, determine whether the solutions will be rational or irrational.
::在不解决等式的情况下,确定解决办法是合理还是不合理。

$\begin{aligned} x^{2} = - 4 x + 20 \end{aligned}$
::x24x+20

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

$\begin{aligned} 3 x^{2} - 11 x = 10 \end{aligned}$
::3x2 - 11x=10

$\begin{aligned} \frac{1}{2} x^{2} + 2 x + \frac{2}{3} = 0 \end{aligned}$
::12x2+2x+23=0

$\begin{aligned} x^{2} - 10 x + 25 = 0 \end{aligned}$
::x2 - 10x+25=0

$\begin{aligned} x^{2} = 5 x \end{aligned}$
::x2=5x (x2=5x)

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

Marty is outside his apartment building. He needs to give his roommate Yolanda her cell phone but he does not have time to run upstairs to the third 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 $\begin{aligned} y = - 32 t^{2} + 55 t + 4 \end{aligned}$ .)
::Marty在公寓楼外,他需要给他的室友Yolanda她的手机,但他没有时间跑到楼上三楼给她手机,他直直往上扔,垂直速度为55英尺/秒。如果她高到36英尺,电话会接通她吗? (提示:高度的方程式是y32t2+55t+4)

Bryson owns a business that manufactures and sells tires. The revenue from selling the tires in the month of July is given by the function $\begin{aligned} R = x (200 - 0.4 x) \end{aligned}$ where $\begin{aligned} x \end{aligned}$ is the number of tires sold. Can Bryson’s business generate revenue of $20,000 in the month of July?
::布赖森拥有一个制造和销售轮胎的企业。 7月份出售轮胎的收入由R=x(200-0.4x)函数提供,x是售出的轮胎数量。布赖森的生意能否在7月份创造20 000美元的收益?

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。

10.10 利用争议解决

Section outline

General

利用争议解决

Solutions Using the Discriminant
::利用争议解决

Find the Discriminant of a Quadratic Equation
::查找二次赤道的偏差

Interpret the Discriminant of a Quadratic Equation
::解释二次等同的辩驳词

Determining the Nature of Solutions
::确定解决办法的性质

Solve Real-World Problems Using Quadratic Functions and Interpreting the Discriminant
::利用四分法函数和解释争议解释解决现实世界问题

Real-World Application: Football
::真实世界应用:足球

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)

Section outline

General

利用争议解决

Solutions Using the Discriminant ::利用争议解决

Find the Discriminant of a Quadratic Equation ::查找二次赤道的偏差

Interpret the Discriminant of a Quadratic Equation ::解释二次等同的辩驳词

Determining the Nature of Solutions ::确定解决办法的性质

Solve Real-World Problems Using Quadratic Functions and Interpreting the Discriminant ::利用四分法函数和解释争议解释解决现实世界问题

Real-World Application: Football ::真实世界应用:足球

Example ::示例示例示例示例

Example 1 ::例1

Review ::回顾

Review (Answers) ::回顾(答复)

Solutions Using the Discriminant
::利用争议解决

Find the Discriminant of a Quadratic Equation
::查找二次赤道的偏差

Interpret the Discriminant of a Quadratic Equation
::解释二次等同的辩驳词

Determining the Nature of Solutions
::确定解决办法的性质

Solve Real-World Problems Using Quadratic Functions and Interpreting the Discriminant
::利用四分法函数和解释争议解释解决现实世界问题

Real-World Application: Football
::真实世界应用:足球

Example
::示例示例示例示例

Example 1
::例1

Review
::回顾

Review (Answers)
::回顾(答复)