节: 使用保理法解析四方 | 5.4 使用保理法解决四方

章节大纲

The height of a ball that is thrown straight up in the air from a height of 2 meters above the ground with a velocity of 9 meters per second is given by the quadratic equation $\begin{aligned} h = - 5 t^{2} + 9 t + 2 \end{aligned}$ , where t is the time in seconds. How long does it take the ball to hit the ground?
::一个球从地面上2米高处直直投到空中,速度为每秒9米的球的高度是由四方方程式 h5t2+9t+2给出的,其时间是几秒。球撞击地面需要多长时间?

Factoring Quadratics
::刻度计乘

To solve word problems involving quadratic equations, you will need to know how to either factor the quadratic equation and set it equal to zero or else use the quadratic formula , in order to solve for $\begin{aligned} x . \end{aligned}$
::要解决涉及二次方程的字问题, 您需要知道如何将二次方程乘以, 并将其设为零, 或者使用二次方程, 才能解决 x 。

Solve the following quadratic by factoring.
::通过保理来解决以下的二次方位。

$\begin{aligned} x^{2} - 9 x + 18 = 0 \end{aligned}$

::x2- 9x+18=0

The only difference between this problem and previous ones is the addition of the $\begin{aligned} = \end{aligned}$ sign. Now that this is present, we need to solve for $\begin{aligned} x \end{aligned}$ . We can still factor the way we always have. Because $\begin{aligned} a = 1 \end{aligned}$
::此问题和先前问题的唯一区别是添加 = 符号。现在, 有了这个符号, 我们需要解决 x。我们还可以考虑我们一直拥有的方式。因为 a=1

$\begin{aligned} x^{2} - 9 x + 18 & = 0 \\ (x - 6) (x - 3) & = 0 \end{aligned}$

::x2- 9x+18=0(x-6)(x-3)=0

Now, we have two factors that, when multiplied, equal zero. Recall that when two numbers are multiplied together and one of them is zero, the product is always zero.
::现在,我们有两个因素, 当乘以时, 等于零。请注意, 当两个数字一起乘以, 一个是零时, 产品总是为零。

Zero-Product Property : If $\begin{aligned} a b = 0 \end{aligned}$ , then $\begin{aligned} a = 0 \end{aligned}$ or $\begin{aligned} b = 0 \end{aligned}$ .
::零财产:如果ab=0,那么a=0或b=0。

This means that $\begin{aligned} x - 6 = 0 \end{aligned}$ OR $\begin{aligned} x - 3 = 0 \end{aligned}$ . Therefore , $\begin{aligned} x = 6 \end{aligned}$ or $\begin{aligned} x = 3 \end{aligned}$ . There will always be the same number of solutions as factors.
::这意味着 x-6=0 OR x-3=0。因此, x=6 或 x=3 将总是有与因数相同数量的解决方案。

Check your answer:
::检查您的答案 :

$\begin{aligned} 6^{2} - 9 (6) + 18 & = 0 o r 3^{2} - 9 (3) + 18 = 0 \\ 36 - 54 + 18 & = 0 9 - 27 + 18 = 0 \end{aligned}$

::62-9(6)+18=0or32-9(3)+18=036-54+18=09-27+18=0

Solve the following quadratic by factoring.
::通过保理来解决以下的二次方位。

$\begin{aligned} 6 x^{2} + x - 4 = 11 \end{aligned}$

::6x2+x-4=11

At first glance, this might not look factorable to you. However, before we factor, we must combine like terms . Also, the Zero-Product Property tells us that in order to solve for the factors, one side of the equation must be zero.
::乍一看,在你们看来,这也许不是因素。然而,在我们考虑之前,我们必须将类似条件结合起来。此外,“零生产财产”告诉我们,为了解决各种因素,等式的一面必须是零。

$\begin{aligned} 6 x^{2} + x - 4 = 11 \\ \underline{- 11 = - 11} \\ 6 x^{2} + x - 15 = 0 \end{aligned}$

::6x2+x-4=11-1111_6x2+x-15=0

Now, factor. The product of $\begin{aligned} a c \end{aligned}$ is -90. What are the two factors of -90 that add up to 1? 10 and -9. Expand the $\begin{aligned} x - \end{aligned}$ term and factor.
::系数。 ac 的产物是 -90。 -90 的这两个系数加起来等于 1? 10 和 -9. 扩大 x- 期数和系数。

$\begin{aligned} 6 x^{2} + x - 15 & = 0 \\ 6 x^{2} - 9 x + 10 x - 15 & = 0 \\ 3 x (2 x - 3) + 5 (2 x - 3) & = 0 \\ (2 x - 3) (3 x + 5) & = 0 \end{aligned}$

::6x2+x-15=06x2-9x+10x-15=03x(2x-3)+5(2x-3)=0(2x-3)(3x+5)=0

Lastly, set each factor equal to zero and solve.
::最后,确定每个因素等于零并解决。

$\begin{aligned} 2 x - 3 & = 0 3 x + 5 = 0 \\ 2 x & = 3 o r 3 x = - 5 \\ x & = \frac{3}{2} x = - \frac{5}{3} \end{aligned}$

::2 - 3=0 3x+5=02x=3or3x5x=32x53

Check your work:
::检查您的工作 :

$\begin{aligned} 6 {(\frac{3}{2})}^{2} + \frac{3}{2} - 4 & = 11 6 {(- \frac{5}{3})}^{2} - \frac{5}{3} - 4 = 11 \\ 6 \cdot \frac{9}{4} + \frac{3}{2} - 4 & = 11 o r 6 \cdot \frac{25}{9} - \frac{5}{3} - 4 = 11 \\ \frac{27}{2} + \frac{3}{2} - 4 & = 11 \frac{50}{3} - \frac{5}{3} - 4 = 11 \\ 15 - 4 & = 11 15 - 4 = 11 \end{aligned}$

::6(32)2+32-4=11 6(-532)+32-4=111 6(-53)-2-53-4-53-4=1 116-1994+32-4=1111 或 62559-53-44=1172+32-4=11503-534=1115-4=1115-4=11

Solve the following quadratic by factoring.
::通过保理来解决以下的二次方位。

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

::10x2 - 25x=0

Here is an example of a quadratic equation without a constant term. The only thing we can do is take out the GCF.
::这是一个没有固定期限的二次等式的例子,我们唯一能做的就是消灭全球合作框架。

$\begin{aligned} 10 x^{2} - 25 x & = 0 \\ 5 x (2 x - 5) & = 0 \end{aligned}$

::10x2-25x=05x(2x-5)=0

Set the two factors equal to zero and solve.
::设定等于零和溶解的两个系数。

$\begin{aligned} 5 x & = 0 2 x - 5 = 0 \\ x & = 0 o r 2 x = 5 \\ x = \frac{5}{2} \end{aligned}$

::5x=02x-5=0x=0x=0or 2x=5x=52

Check:
::检查 :

$\begin{aligned} 10 (0)^{2} - 25 (0) = 0 10 {(\frac{5}{2})}^{2} - 25 (\frac{5}{2}) = 0 \\ 0 = 0 o r 10 \cdot \frac{25}{4} - \frac{125}{2} = 0 \\ \frac{125}{2} - \frac{125}{2} = 0 \end{aligned}$

::10(0)2-25(0)=010(52)2-25(52)=00=0=0or10_254-1252=01252-1252=0

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the time it takes the ball to hit the ground.
::早些时候,有人要求你找到球击中地面所需的时间。

When the ball hits the ground, the height h is 0. So the equation becomes $\begin{aligned} 0 = - 5 t^{2} + 9 t + 2 \end{aligned}$
::当球击中地面时,高度 h 是 0。所以方程式会变成 05t2+9t+2

Let's factor and solve for t . $\begin{aligned} - 5 t^{2} + 9 t + 2 \end{aligned}$
::让我们为 t. - 5t2+9t+2 的因数和解析

We need to find the factors of $\begin{aligned} - 10 \end{aligned}$ that add up to 9. Testing the possibilities, we find 10 and -1 to be the correct combination.
::我们需要找到 10 的因素共合9. 测试可能性, 我们发现10 和 -1 是正确的组合。

$\begin{aligned} - 5 t^{2} + 10 t - t + 2 \end{aligned}$ = $\begin{aligned} (- 5 t^{2} + 10 t) + (- t + 2) \end{aligned}$ = $\begin{aligned} 5 t (- t + 2) + (- t + 2) \end{aligned}$ = $\begin{aligned} (5 t + 1) (- t + 2) \end{aligned}$
::-5t2+10t+10t+2=(-5t2+10t)+(-T+2)=5t(-T+2)+(-T+2)+(-T+2)+(-T+2)=5t+1(-T+2)=5t+1(-T+2)

Now set this factorization equal to zero and solve.
::现在将此乘数设置为零, 并解决。

$\begin{aligned} (5 t + 1) (- t + 2) = 0 \end{aligned}$
:5t+1)(-t+2)=0

Because t represents the time, it must be positive. Only $\begin{aligned} (- t + 2) = 0 \end{aligned}$ results in a positive value.
::t 表示时间, 必须是正值。只有 (- t+2) =0 才会产生正值。

$\begin{aligned} t = 2 \end{aligned}$ , therefore it takes the ball 2 seconds to reach the ground.
::t=2,所以球需要2秒到达地面。

Example 2
::例2

Solve the following equation by factoring.
::通过乘法解决以下方程。

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

$\begin{aligned} a c = 36 \end{aligned}$ . The factors of 36 that also add up to -12 are -6 and -6. Expand the $\begin{aligned} x - \end{aligned}$ term and factor.
::ac=36. 与 -12相加的36个系数的乘数为 -6和 -6. 扩大x-期和因数。

$\begin{aligned} 4 x^{2} - 12 x + 9 & = 0 \\ 4 x^{2} - 6 x - 6 x + 9 & = 0 \\ 2 x (2 x - 3) - 3 (2 x - 3) & = 0 \\ (2 x - 3) (2 x - 3) & = 0 \end{aligned}$

::4x2 - 12x+9=04x2 - 6x- 6x+9=02x(2x-3) - 3(2x-3)=0(2x-3)(2x-3)=0

The factors are the same. When factoring a perfect square trinomial , the factors will always be the same. In this instance, the solutions for $\begin{aligned} x \end{aligned}$ will also be the same. Solve for $\begin{aligned} x \end{aligned}$ .
::因素是相同的。当将一个完全平方的三角因素考虑在内时, 因素总是相同的。在此情况下, x 的解决方案也是相同的。解决 x 。

$\begin{aligned} 2 x - 3 & = 0 \\ 2 x & = 3 \\ x & = \frac{3}{2} \end{aligned}$

::2x-3=02x=3x=32

When the two factors are the same, we call the solution for $\begin{aligned} x \end{aligned}$ a double root because it is the solution twice.
::当这两个因素相同时,我们称x的解决方案为双根,因为它是两次解决方案。

Example 3
::例3

Solve the following equation by factoring.
::通过乘法解决以下方程。

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

Here, we need to get everything on the same side of the equals sign in order to factor.
::在这里,我们需要把所有东西都放在平等标志的同一边,以便考虑。

$\begin{aligned} x^{2} - 5 x & = 6 \\ x^{2} - 5 x - 6 & = 0 \end{aligned}$

::x2 - 5x=6x2 - 5x-6=0

Because there is no number in front of $\begin{aligned} x^{2} \end{aligned}$ , we need to find the factors of -6 that add up to -5.
::因为x2前面没有数字, 我们需要找到 -6 的因素, 加到 -5 。

$\begin{aligned} (x - 6) (x + 1) = 0 \end{aligned}$

:x-6)(x+1)=0

Solving each factor for $\begin{aligned} x \end{aligned}$ , we get that $\begin{aligned} x = 6 \end{aligned}$ or $\begin{aligned} x = - 1 \end{aligned}$ .
::正在解析 x 的每个因数时, 我们得到 x=6 或 x=1 。

Example 4
::例4

Solve the following equation by factoring.
::通过乘法解决以下方程。

$\begin{aligned} 8 x - 20 x^{2} = 0 \end{aligned}$
::8 - 20x2=0

Here there is no constant term. Find the GCF to factor.
::这里没有固定的术语。找出绿色气候基金的因素。

$\begin{aligned} 8 x - 20 x^{2} & = 0 \\ 4 x (2 - 5 x) & = 0 \end{aligned}$

::8 - 20x2=04x(2 - 5x)=0

Solve each factor for $\begin{aligned} x \end{aligned}$ .
::解决 x 的每个因数。

$\begin{aligned} 4 x & = 0 2 - 5 x = 0 \\ x & = 0 o r 2 = 5 x \\ \frac{2}{5} = x \end{aligned}$

::4x=02-5x=0x=0x=0or 2=5x25=x

Example 5
::例5

Solve the following equation by factoring.
::通过乘法解决以下方程。

$\begin{aligned} 12 x^{2} + 13 x + 7 = 12 - 4 x \end{aligned}$
::12x2+13x+7=12-4x

This problem is slightly more complicated. Combine all like terms onto the same side of the equals sign so that one side is zero.
::这个问题略为复杂。将相同条件合并到等值符号的同一面, 使一面为零。

$\begin{aligned} 12 x^{2} + 13 x + 7 & = 12 - 4 x \\ 12 x^{2} + 17 x - 5 & = 0 \end{aligned}$

::12x2+13x+7=12-4x12x2+17x-5=0

$\begin{aligned} a c = - 60 \end{aligned}$ . The factors of -60 that add up to 17 are 20 and -3. Expand the $\begin{aligned} x - \end{aligned}$ term and factor.
::ac60. 总计17个的-60系数为20和-3。扩大x-期和系数。

$\begin{aligned} 12 x^{2} + 17 x - 5 & = 0 \\ 12 x^{2} + 20 x - 3 x - 5 & = 0 \\ 4 x (3 x + 5) - 1 (3 x + 5) & = 0 \\ (3 x + 5) (4 x - 1) & = 0 \end{aligned}$

::12x2+17x-5=012x2+20x-3x-5=04x(3x+5)-1(3x+5)=0(3x+5)-4x-1=0

Solve each factor for $\begin{aligned} x \end{aligned}$ .
::解决 x 的每个因数。

$\begin{aligned} 3 x + 5 & = 0 4 x - 1 = 0 \\ 3 x & = - 5 o r 4 x = 1 \\ x & = - \frac{5}{3} x = \frac{1}{4} \end{aligned}$

::3x+5=04x-1=03x=03x=%5or4x=1x*5x=14

Review
::回顾

Solve the following quadratic equations by factoring, if possible.
::如果可能的话,通过乘数来解决以下四方方程式。

$\begin{aligned} x^{2} + 8 x - 9 = 0 \end{aligned}$
::x2+8x-9=0

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

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

$\begin{aligned} 12 x^{2} + 7 x - 10 = 0 \end{aligned}$
::12x2+7x-10=0

$\begin{aligned} x^{2} = 9 \end{aligned}$
::x2=9

$\begin{aligned} 30 x + 25 = - 9 x^{2} \end{aligned}$
::30x+259x2

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

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

$\begin{aligned} 3 x^{2} + 28 x = - 32 \end{aligned}$
::3x2+28x%32

$\begin{aligned} 36 x^{2} - 48 = 1 \end{aligned}$
::36x2-48=1

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

$\begin{aligned} 5 x^{2} + 12 x + 4 = 0 \end{aligned}$
::5x2+12x+4=0

Challenge Solve these quadratic equations by factoring. They are all factorable.
::通过保理来解决这些二次方程式。它们都是可以保理的。

$\begin{aligned} 8 x^{2} + 8 x - 5 = 10 - 6 x \end{aligned}$
::8x2+8x-5=10-6x

$\begin{aligned} - 18 x^{2} = 48 x + 14 \end{aligned}$
::- 18x2=48x+14

$\begin{aligned} 36 x^{2} - 24 = 96 x - 39 \end{aligned}$
::36x2--24=96x-39

Real Life Application George is helping his dad build a fence for the backyard. The total area of their backyard is 1600 square feet. The width of the house is half the length of the yard, plus 7 feet. How much fencing does George’s dad need to buy?
::现实生活应用程序乔治正在帮助他的爸爸为后院修建围栏。他们后院的总面积是1600平方英尺。房子宽度是院子长度的一半,加上7英尺。乔治的父亲需要买多少围栏?

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

Factoring Quadratics ::刻度计乘

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)

Factoring Quadratics
::刻度计乘

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)