Course: 9.11 利用完美广场三角法进行保理

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使用完美广场三边化法进行分化
Factorization using Perfect Square Trinomials
::使用完美广场三边化法进行分化

We use the square of a binomial formula to factor perfect square trinomials. A perfect square trinomial has the form $\begin{aligned} a^{2} + 2 a b + b^{2} \end{aligned}$ or $\begin{aligned} a^{2} - 2 a b + b^{2} \end{aligned}$ .
::我们使用二进制公式的正方形来乘以完美的平方三角。完美的平方三角形有 A2+2ab+b2 或 a2- 2ab+b2 的形式。

In these special kinds of trinomials, the first and last terms are perfect squares and the middle term is twice the product of the square roots of the first and last terms. In a case like this, the polynomial factors into perfect squares:
::在这些特殊的三角关系中,第一个和最后一个条件都是完美的正方形,中期是第一个和最后一个条件的平方根的两倍。

$\begin{aligned} a^{2} + 2 a b + b^{2} & = (a + b)^{2} \\ a^{2} - 2 a b + b^{2} & = (a - b)^{2} \end{aligned}$

::a2+2ab+b2=(a+b)2a2-2a2-2ab+b2=(a-b)2

Once again, the key is figuring out what the $\begin{aligned} a \end{aligned}$ and $\begin{aligned} b \end{aligned}$ terms are.
::再次,关键是要弄清楚什么是a和b的术语。

Factoring Perfect Square Trinomials
::计算完美的广场三角架

1. Factor the following perfect square trinomials:
::1. 将下列完全正方形三角体乘以:

a) $\begin{aligned} x^{2} + 8 x + 16 \end{aligned}$
::a) x2+8x+16

The first step is to recognize that this expression is a perfect square trinomial.
::第一步是承认这种表达方式是完全平坦的三重用语。

First, we can see that the first term and the last term are perfect squares. We can rewrite $\begin{aligned} x^{2} + 8 x + 16 \end{aligned}$ as $\begin{aligned} x^{2} + 8 x + 4^{2} \end{aligned}$ .
::首先,我们可以看到第一个学期和最后一个学期是完美的方形。我们可以将 x2+8x+16 重写为 x2+8x+42 。

Next, we check that the middle term is twice the product of the square roots of the first and the last terms. This is true also since we can rewrite $\begin{aligned} x^{2} + 8 x + 16 \end{aligned}$ as $\begin{aligned} x^{2} + 2 \cdot 4 \cdot x + 4^{2} \end{aligned}$ .
::接下来,我们检查中期是第一个条件和最后一个条件的平方根的产物的两倍。这也是由于我们可以将 x2+8x+16 重写为 x2+2+24x+42 。

This means we can factor $\begin{aligned} x^{2} + 8 x + 16 \end{aligned}$ as $\begin{aligned} (x + 4)^{2} \end{aligned}$ . We can check to see if this is correct by multiplying $\begin{aligned} (x + 4)^{2} = (x + 4) (x + 4) \end{aligned}$ :
::这意味着我们可以将乘数 x2+8x+16 改为 (x+4) 2。我们可以通过乘法( x+4) 2=( x+4) (x+4) 来检查是否正确 :

$\begin{aligned} x + 4 \\ \underline{x + 4} \\ 4 x + 16 \\ \underline{x^{2} + 4 x} \\ x^{2} + 8 x + 16 \end{aligned}$

::x+4x+4_4x+16x2+4x_x2+8x16

The answer checks out.
::答案检查出来。

Note: We could factor this trinomial without recognizing it as a perfect square. We know that a trinomial factors as a product of two binomials:
::注意:我们可以在不承认三重正方形为完美正方形的情况下将三重三重三重三重一乘。我们知道三重三重一因素是两重二重一的产物:

$\begin{aligned} (x) (x) \end{aligned}$

:x)(x)

We need to find two numbers that multiply to 16 and add to 8. We can write 16 as the following products:
::我们需要找到两个乘以16的号码,再加到8。我们可以把16写成以下产品:

$\begin{aligned} 16 = 1 \cdot 16 & and & 1 + 16 = 17 \\ 16 = 2 \cdot 8 & and & 2 + 8 = 10 \\ 16 = 4 \cdot 4 & and & 4 + 4 = 8 T h e s e a r e t h e c o r r e c t n u m b e r s \end{aligned}$

::16=116和1+16=1716=28和2+8=1016=44和4+4=8

So we can factor $\begin{aligned} x^{2} + 8 x + 16 \end{aligned}$ as $\begin{aligned} (x + 4) (x + 4) \end{aligned}$ , which is the same as $\begin{aligned} (x + 4)^{2} \end{aligned}$ .
::因此我们可以将乘数x2+8x+16作为(x+4)(x+4),这与(x+4)2相同。

Once again, you can factor perfect square trinomials the normal way, but recognizing them as perfect squares gives you a useful shortcut.
::再一次,你可以以正常的方式将完美的正方形三角乘以,但承认它们是完美的正方形,这给了你一条有用的捷径。

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

Rewrite $\begin{aligned} x^{2} + 4 x + 4 \end{aligned}$ as $\begin{aligned} x^{2} + 2 \cdot (- 2) \cdot x + (- 2)^{2} \end{aligned}$ .
::重写 x2+4x+4 为 x2+2(-2)x+(-2)2。

We notice that this is a perfect square trinomial, so we can factor it as $\begin{aligned} (x - 2)^{2} \end{aligned}$ .
::我们注意到这是一个完美的三重正方形, 所以我们可以把它作为(x-2)2乘以(x-2)2。

c) $\begin{aligned} x^{2} + 14 x + 49 \end{aligned}$
:c) x2+14x+49

Rewrite $\begin{aligned} x^{2} + 14 x + 49 \end{aligned}$ as $\begin{aligned} x^{2} + 2 \cdot 7 \cdot x + 7^{2} \end{aligned}$ .
::重写 x2+14x+49为 x2+2%7x+72。

We notice that this is a perfect square trinomial, so we can factor it as $\begin{aligned} (x + 7)^{2} \end{aligned}$ .
::我们注意到,这是一个完全平方的三角关系,因此我们可以将其乘以(x+7)2。

2. Factor the following perfect square trinomials:
::2. 将下列完全正方形三角体乘以:

a) $\begin{aligned} 4 x^{2} + 20 x + 25 \end{aligned}$
:a) 4x2+20x+25

Rewrite $\begin{aligned} 4 x^{2} + 20 x + 25 \end{aligned}$ as $\begin{aligned} (2 x)^{2} + 2 \cdot 5 \cdot (2 x) + 5^{2} \end{aligned}$ .
::将 4x2+20x+25 重写为(2x)2+25}(2x)+52。

We notice that this is a perfect square trinomial and we can factor it as $\begin{aligned} (2 x + 5)^{2} \end{aligned}$ .
::我们注意到,这是一个完美的三重正方形,我们可以将其乘以(2x+5)2。

b) $\begin{aligned} 9 x^{2} - 24 x + 16 \end{aligned}$
::b) 9x2-24x+16

Rewrite $\begin{aligned} 9 x^{2} - 24 x + 16 \end{aligned}$ as $\begin{aligned} (3 x)^{2} + 2 \cdot (- 4) \cdot (3 x) + (- 4)^{2} \end{aligned}$ .
::将 9x2- 24x+16 重写为( 3x) 2+2}( 4) ( 3x) +( 4) 2 。

We notice that this is a perfect square trinomial and we can factor it as $\begin{aligned} (3 x - 4)^{2} \end{aligned}$ .
::我们注意到,这是一个完美的三重正方形,我们可以将其乘以(3x-4)2。

We can check to see if this is correct by multiplying $\begin{aligned} (3 x - 4)^{2} = (3 x - 4) (3 x - 4) \end{aligned}$ :
::我们可以通过乘法( 3x- 4) 2=( 3x- 4) (3x- 4) (3x- 4) 来检查是否正确 :

$\begin{aligned} 3 x - 4 \\ \underline{3 x - 4} \\ - 12 x + 16 \\ \underline{9 x^{2} - 12 x} \\ 9 x^{2} - 24 x + 16 \end{aligned}$

::3x - 43x - 4_ - 12x+169x2 - 12x_ 9x2 - 24x+16

The answer checks out.
::答案检查出来。

c) $\begin{aligned} x^{2} + 2 x y + y^{2} \end{aligned}$
::x2+2xy+y2 (c) x2+2xy+y2

$\begin{aligned} x^{2} + 2 x y + y^{2} \end{aligned}$
::x2+2xy+y2

We notice that this is a perfect square trinomial and we can factor it as $\begin{aligned} (x + y)^{2} \end{aligned}$ .
::我们注意到,这是一个完美的三重正方形,我们可以将它作为(x+y)2 乘以(x+y)2。

Solve Quadratic Polynomial Equations by Factoring
::以保理方式解决二次二次二次等同等量

With the methods we’ve learned in the last two sections, we can factor many kinds of quadratic polynomials . This is very helpful when we want to solve them. Remember the process we learned earlier:
::以我们在最后两节中学到的方法,我们可以将多种四边形多面体考虑进去。当我们想解决这些问题时,这非常有用。记住我们早先学到的过程:

If necessary, rewrite the equation in standard form so that the right-hand side equals zero.
::如有必要,以标准格式重写方程式,使右侧等于零。

Factor the polynomial completely.
::将复数完全乘以。

Use the zero-product rule to set each factor equal to zero.
::使用零产品规则将每个系数设定为零。

Solve each equation from step 3.
::从第3步解决每个方程式

Check your answers by substituting your solutions into the original equation
::通过将您的解决方案替换为原始方程来检查您的答案

We can use this process to solve quadratic polynomials using the factoring methods we just learned.
::我们可以利用这个过程用我们刚刚学到的计数方法解决四边多面性多边性。

Solving for Unknown Values
::解决未知值

Solve the following polynomial equations.
::解决以下多面方程式。

a) $\begin{aligned} x^{2} + 12 x + 36 = 0 \end{aligned}$
::a) x2+12x+36=0

Rewrite:
::重写 :

The equation is already in the correct form.
::方程已经以正确形式出现。

Factor:
::因素 :

Rewrite $\begin{aligned} x^{2} + 12 x + 36 = 0 \end{aligned}$ as $\begin{aligned} x^{2} + 2 (6 x) + 6^{2} = 0 \end{aligned}$ . We notice that this is a perfect square trinomial and we can factor it as $\begin{aligned} (x + 6)^{2} \end{aligned}$ .
::将 x2+12x+36=0 重写为 x2+2(6x)+62=0。我们注意到这是一个完美的正方方形三角, 我们可以将其乘以 (x+6) 2 。

Set the factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x + 6 = 0 \end{aligned}$

::x+6=0 x+6=0

Solve:
::解决 :

$\begin{aligned} \underline{\underline{x = - 6}} \end{aligned}$

::6______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Check: Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} (- 6)^{2} + 12 (- 6) + 36 = & Substitute in -6. \\ 36 + - 72 + 36 = & Simplify. \\ 72 + - 72 = 0 & Checks out. \end{aligned}$

:-6)2+12(-6)+36= -6.3672+36=简化.7272=0检查退出。)

b) $\begin{aligned} x^{2} - 24 x = - 144 \end{aligned}$
:b) x2-24x144

Rewrite: $\begin{aligned} x^{2} - 24 x = - 144 \end{aligned}$ is rewritten as $\begin{aligned} x^{2} - 24 x + 144 = 0 \end{aligned}$
::重写: x2 - 24x\\\\\ 144 被重写为 x2 - 24x+144=0

Factor:
::因素 :

$\begin{aligned} x^{2} - 24 x + 144 = x^{2} + 2 (- 12) x + (- 12)^{2} = (x - 12)^{2} \end{aligned}$
::x2 - 24x+144=x2+2(- 12x+(- 12)2=(x- 12)2

Set the factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x - 12 = 0 \end{aligned}$

::x- 12=0

Solve:
::解决 :

$\begin{aligned} \underline{\underline{x = 12}} \end{aligned}$

::x=12__

Check: Substitute the solution back into the original equation.
::复选: 将溶液替换回原始方程式。

$\begin{aligned} (12)^{2} - 24 (12) + 144 = & Substitute in 12. \\ 144 - 288 + 144 = & Simplify. \\ 288 - 288 = 0 & Checks out. \end{aligned}$

:12)2-24(12)+144=12.144-288+144=简化288-288=0

Examples
::实例

Solve the following polynomial equations:
::解决下列多元方程式:

Example 1
::例1

$\begin{aligned} x^{2} + x + 0.25 = 0 \end{aligned}$
::x2+x+0.25=0

$\begin{aligned} x^{2} + x + 0.25 = 0 \end{aligned}$
::x2+x+0.25=0

Rewrite: The equation is in the correct form already.
::重写: 方程式已经处于正确的形式。

Factor: Rewrite $\begin{aligned} x^{2} + x + 0.25 = 0 \end{aligned}$ as $\begin{aligned} x^{2} + 2 \cdot (0.5) x + (0.5)^{2} \end{aligned}$ .
::系数: 重写 x2+x+0. 25=0 x2+2( 0. 5) x+( 0. 5) 2。

We recognize this as a perfect square. This factors as $\begin{aligned} (x + 0.5)^{2} = 0 \end{aligned}$ or $\begin{aligned} (x + 0.5) (x + 0.5) = 0 \end{aligned}$
::我们承认这是一个完美的平方。这些系数是(x+0.5)2=0或(x+0.5)(x+0.5)=0。

Set the factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x + 0.25 = 0 \end{aligned}$

::x+0.25=0

Solve:
::解决 :

$\begin{aligned} \underline{\underline{x = - 0.5}} \end{aligned}$

::0.5___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Check: Substitute the solution back into the original equation.
::复选: 将溶液替换回原始方程式。

$\begin{aligned} (- 0.5)^{2} + - 0.5 + 0.25 & = Substitute in -0.5 \\ 0.25 + - 0.5 + 0.25 & = Simplify. \\ 0.5 - 0.5 & = 0 Checks out. \end{aligned}$

:-0.5)20.5+0.25=0.50.250.5+0.25=简化.0.5-0.5=0检查退出。

Example 2
::例2

$\begin{aligned} x^{2} - 81 = 0 \end{aligned}$
::x2 - 81=0

$\begin{aligned} x^{2} - 81 = 0 \end{aligned}$
::x2 - 81=0

Rewrite: this is not necessary since the equation is in the correct form already
::重写: 不需要重写: 因为方程已经以正确形式出现

Factor: Rewrite $\begin{aligned} x^{2} - 81 \end{aligned}$ as $\begin{aligned} x^{2} - 9^{2} \end{aligned}$ .
::系数:重写 x2-81为 x2-92。

We recognize this as a difference of squares . This factors as $\begin{aligned} (x - 9) (x + 9) = 0 \end{aligned}$ .
::我们认识到这是平方的差别,这些因素是(x-9)(x+9)=0。

Set each factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x - 9 = 0 & or & x + 9 = 0 \end{aligned}$

::x- 9=0orx+9=0

Solve:
::解决 :

$\begin{aligned} \underline{\underline{x = 9}} & or & \underline{\underline{x = - 9}} \end{aligned}$

::x=9__orx9__

Check: Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} x = 9 & 9^{2} - 81 = 81 - 81 = 0 & checks out \\ x = - 9 & (- 9)^{2} - 81 = 81 - 81 = 0 & checks out \end{aligned}$

::x=992- 81=81- 81=0 checks outx%9(- 9)2- 81=81- 81=0 checks out

c) $\begin{aligned} x^{2} + 20 x + 100 = 0 \end{aligned}$
::c) x2+20x+100=0

Rewrite: this is not necessary since the equation is in the correct form already
::重写: 不需要重写: 因为方程已经以正确形式出现

Factor: Rewrite $\begin{aligned} x^{2} + 20 x + 100 \end{aligned}$ as $\begin{aligned} x^{2} + 2 \cdot 10 \cdot x + 10^{2} \end{aligned}$ .
::系数: 重写 x2+20x+100为 x2+2%10x+102。

We recognize this as a perfect square. This factors as $\begin{aligned} (x + 10)^{2} = 0 \end{aligned}$ or $\begin{aligned} (x + 10) (x + 10) = 0 \end{aligned}$
::我们认识到这是一个完美的平方。这些系数是 (x+10) 2=0 或 (x+10) (x+10) (x+10)=0。

Set each factor equal to zero:
::设定等于零的因数 :

$\begin{aligned} x + 10 = 0 & or & x + 10 = 0 \end{aligned}$

::x+10=0orx+10=0

Solve:
::解决 :

$\begin{aligned} \underline{\underline{x = - 10}} & or & \underline{\underline{x = - 10}} This is a double root. \end{aligned}$

::x10_orx_Q_10_这是一个双根根。

Check: Substitute each solution back into the original equation.
::复选: 将每个溶液替换回原始方程。

$\begin{aligned} x = 10 & (- 10)^{2} + 20 (- 10) + 100 = 100 - 200 + 100 = 0 & checks out \end{aligned}$

::x=10(-10)2+20(-10)+100=100-200+100=100=0检查

Review
::回顾

Factor the following perfect square trinomials.
::乘以以下完美的平方三角曲线

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

$\begin{aligned} x^{2} - 18 x + 81 \end{aligned}$
::x2 - 18x+81

$\begin{aligned} - x^{2} + 24 x - 144 \end{aligned}$
::-x2+24x-144

$\begin{aligned} x^{2} + 14 x + 49 \end{aligned}$
::x2+14x+49

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

$\begin{aligned} 25 x^{2} + 60 x + 36 \end{aligned}$
::25x2+60x+36

$\begin{aligned} 4 x^{2} - 12 x y + 9 y^{2} \end{aligned}$
::4x2 - 12xy+9y2

$\begin{aligned} x^{4} + 22 x^{2} + 121 \end{aligned}$
::x4+22x2+121

Solve the following quadratic equations using factoring.
::使用保理系数解决以下二次方程。

$\begin{aligned} x^{2} - 11 x + 30 = 0 \end{aligned}$
::x2 - 11x+30=0

$\begin{aligned} x^{2} + 4 x = 21 \end{aligned}$
::x2+4x=21

$\begin{aligned} x^{2} + 49 = 14 x \end{aligned}$
::x2+49=14x

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

$\begin{aligned} x^{2} - 24 x + 144 = 0 \end{aligned}$
::x2 - 24x+144=0

$\begin{aligned} 4 x^{2} - 25 = 0 \end{aligned}$
::4x2-25=0

$\begin{aligned} x^{2} + 26 x = - 169 \end{aligned}$
::x2+26x169

$\begin{aligned} - x^{2} - 16 x - 60 = 0 \end{aligned}$
::-x2-16x-60=0

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。

Section outline

General

使用完美广场三边化法进行分化

Factorization using Perfect Square Trinomials ::使用完美广场三边化法进行分化

Factoring Perfect Square Trinomials ::计算完美的广场三角架

Solve Quadratic Polynomial Equations by Factoring ::以保理方式解决二次二次二次等同等量

Solving for Unknown Values ::解决未知值

Examples ::实例

Example 1 ::例1

x 2 + x + 0.25 = 0 ::x2+x+0.25=0

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Factorization using Perfect Square Trinomials
::使用完美广场三边化法进行分化

Factoring Perfect Square Trinomials
::计算完美的广场三角架

Solve Quadratic Polynomial Equations by Factoring
::以保理方式解决二次二次二次等同等量

Solving for Unknown Values
::解决未知值

Examples
::实例

Example 1
::例1

$\begin{aligned} x^{2} + x + 0.25 = 0 \end{aligned}$
::x2+x+0.25=0

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)