Section: 当领导系数不等于1时完成广场 | 5.12 当领导系数不等于1时完成广场

Section outline

The area of a parallelogram is given by the equation $\begin{aligned} 3 x^{2} + 9 x - 5 = 0 \end{aligned}$ , where x is the length of the base. What is the length of this base?
::3x2+9x-5=0 方程式给出了平行图区域, x 是基数的长度。此基数的长度是多少 ?

Completing the Square
::完成广场

When there is a number in front of $\begin{aligned} x^{2} \end{aligned}$ , it will make a little more complicated.
::当 x2 前面有一个数字时, 它会变得更复杂一些。

Let's determine the number c that completes the square of $\begin{aligned} 2 x^{2} - 8 x + c \end{aligned}$ .
::让我们来决定完成 2x2 -8x+c 平方的 c 数字。

Previously , we just added $\begin{aligned} {(\frac{b}{2})}^{2} \end{aligned}$ , but that was when $\begin{aligned} a = 1 \end{aligned}$ . Now that $\begin{aligned} a \neq 1 \end{aligned}$ , we have to take the value of a into consideration. Let's pull out the GCF of 2 and 8 first.
::之前,我们刚刚添加了(b2)2,但当时A=1,现在A=1,我们必须考虑一个价值。让我们首先拿出2和8的绿色气候基金。

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

Now, there is no number in front of $\begin{aligned} x^{2} \end{aligned}$ .
::现在, x2 前面没有数字。

$\begin{aligned} {(\frac{b}{2})}^{2} = {(\frac{4}{2})}^{2} = 4 \end{aligned}$ .
:b2)2=(42)2=4。

Add this number inside the parenthesis and distribute the 2.
::在括号中添加此数字, 并分发 2 。

$\begin{aligned} 2 (x^{2} - 4 x + 4) = 2 x^{2} - 4 x + 8 \end{aligned}$
::2(x2-4x+4)=2x2-4x+8

So, $\begin{aligned} c = 8 \end{aligned}$ .
::所以,c=8。。。

Solve the following by completing the square.
::完成广场,解决以下问题。

$\begin{aligned} 3 x^{2} - 9 x + 11 = 0 \end{aligned}$

::3x2-9x+11=0

Step 1: Write the polynomial so that $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} x \end{aligned}$ are on the left side of the equation and the constants on the right.
::步骤 1: 写入多数值, 使 x2 和 x 位于方程的左侧, 且位于右侧的常数。

$\begin{aligned} 3 x^{2} - 9 x = - 11 \end{aligned}$

::3x2-9x11

Step 2: Pull out $\begin{aligned} a \end{aligned}$ from everything on the left side. Even if $\begin{aligned} b \end{aligned}$ is not divisible by $\begin{aligned} a \end{aligned}$ , the coefficient of $\begin{aligned} x^{2} \end{aligned}$ needs to be 1 in order to complete the square.
::第2步:从左侧的所有东西中抽出一个。即使 b 无法被 a 变异, 但为填平方形, x2 的系数必须为 1 。

$\begin{aligned} 3 (x^{2} - 3 x + \underline{}) = - 11 \end{aligned}$

::3(x2 - 3x) 11

Step 3: Now, complete the square. Determine what number would make a perfect square trinomial .
::第3步:现在,完成方形。决定哪个数字可以使一个完美的方形三角形。

To do this, divide the $\begin{aligned} x - \end{aligned}$ term by 2 and square that number, or $\begin{aligned} {(\frac{b}{2})}^{2} \end{aligned}$ .
::为此,将x-期除以2和该数字的正方形,或(b2)2。

$\begin{aligned} {(\frac{b}{2})}^{2} = {(\frac{3}{2})}^{2} = \frac{9}{4} \end{aligned}$

:b2)2=(32)2=94

Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add $\begin{aligned} a \cdot {(\frac{b}{2})}^{2} \end{aligned}$ to keep the equation balanced.
::第4步:在左侧括号的内部添加此数字。在右侧, 您需要添加 a(b2) 2 来保持方程平衡。

$\begin{aligned} 3 (x^{2} - 3 x + \frac{9}{4}) = - 11 + \frac{27}{4} \end{aligned}$

::3(x2-3x+94)11+274

Step 5: Factor the left side and simplify the right.
::第5步:以左侧为因子,简化右侧。

$\begin{aligned} 3 {(x - \frac{3}{2})}^{2} = - \frac{17}{4} \end{aligned}$

::3(x-32)2174

Step 6: Solve by using square roots.
::步骤6:用平方根解决。

$\begin{aligned} {(x - \frac{3}{2})}^{2} & = - \frac{17}{12} \\ x - \frac{3}{2} & = \pm \frac{i \sqrt{17}}{2 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ x & = \frac{3}{2} \pm \frac{\sqrt{51}}{6} i \end{aligned}$

:x-32)%1712x-32i1723=33x=32×516i

Be careful with the addition of Step 2 and the changes made to Step 4. A very common mistake is to add $\begin{aligned} {(\frac{b}{2})}^{2} \end{aligned}$ to both sides, without multiplying by $\begin{aligned} a \end{aligned}$ for the right side.
::当心增加第2步和修改第4步,一个非常常见的错误是向双方增加(b2)2,而不为右方乘以乘法。

Solve the following by completing the square.
::完成广场,解决以下问题。

$\begin{aligned} 4 x^{2} + 7 x - 18 = 0 \end{aligned}$

::4x2+7x-18=0

Let’s follow the steps from problem #1 above.
::让我们从上面问题1的步子跟上。

Step 1: Write the polynomial so that $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} x \end{aligned}$ are on the left side of the equation and the constants on the right.
::步骤 1: 写入多数值, 使 x2 和 x 位于方程的左侧, 且位于右侧的常数。

$\begin{aligned} 4 x^{2} - 7 x = 18 \end{aligned}$

::4x2-7x=18

Step 2: Pull out $\begin{aligned} a \end{aligned}$ from everything on the left side.
::步骤2:从左侧的一切中抽出一个。

$\begin{aligned} 4 (x^{2} + \frac{7}{4} x + \underline{}) = 18 \end{aligned}$

::4( x2+74x) = 18

Step 3: Now, complete the square. Find $\begin{aligned} {(\frac{b}{2})}^{2} \end{aligned}$ .
::第三步:现在,完成方形。查找 (b2) 2。

$\begin{aligned} {(\frac{b}{2})}^{2} = {(\frac{7}{8})}^{2} = \frac{49}{64} \end{aligned}$

:b2)2=(782=4964)

Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add $\begin{aligned} a \cdot {(\frac{b}{2})}^{2} \end{aligned}$ to keep the equation balanced.
::第4步:在左侧括号的内部添加此数字。在右侧, 您需要添加 a(b2) 2 来保持方程平衡。

$\begin{aligned} 4 (x^{2} + \frac{7}{4} x + \frac{49}{64}) = 18 + \frac{49}{16} \end{aligned}$

::4 (x2+74x+4964)=18+4916

Step 5: Factor the left side and simplify the right.
::第5步:以左侧为因子,简化右侧。

$\begin{aligned} 4 {(x + \frac{7}{8})}^{2} = \frac{337}{16} \end{aligned}$

::4 (x+782) = 33716

Step 6: Solve by using square roots.
::步骤6:用平方根解决。

$\begin{aligned} {(x + \frac{7}{8})}^{2} & = \frac{337}{64} \\ x + \frac{7}{8} & = \pm \frac{\sqrt{337}}{8} \\ x & = - \frac{7}{8} \pm \frac{\sqrt{337}}{8} \end{aligned}$

:x+782=33764x+783378x_78}(x+782=33764x+78}(x+782=33764x+78})

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the length of the base of the parallelogram.
::早些时候,有人要求你找到平行图的底部长度。

We can't factor $\begin{aligned} 3 x^{2} + 9 x - 5 = 0 \end{aligned}$ , so let's follow the step-by-step process we learned in this lesson.
::我们不能乘以 3x2+9x-5=0, 所以让我们遵循我们从这个教训中学到的一步步过程。

Step 1: Write the polynomial so that $\begin{aligned} x^{2} \end{aligned}$ and $\begin{aligned} x \end{aligned}$ are on the left side of the equation and the constants on the right.
::步骤 1: 写入多数值, 使 x2 和 x 位于方程的左侧, 且位于右侧的常数。

$\begin{aligned} 3 x^{2} + 9 x = 5 \end{aligned}$

::3x2+9x=5

Step 2: Pull out $\begin{aligned} a \end{aligned}$ from everything on the left side.
::步骤2:从左侧的一切中抽出一个。

$\begin{aligned} 3 (x^{2} + 3 x + \underline{}) = 5 \end{aligned}$

::3(x2+3x)=5

Step 3: Now, complete the square. Find $\begin{aligned} {(\frac{b}{2})}^{2} \end{aligned}$ .
::第三步:现在,完成方形。查找 (b2) 2。

$\begin{aligned} {(\frac{b}{2})}^{2} = {(\frac{3}{2})}^{2} = \frac{9}{4} \end{aligned}$

:b2)2=(32)2=94

Step 4: Add this number to the interior of the parenthesis on the left side. On the right side, you will need to add $\begin{aligned} a \cdot {(\frac{b}{2})}^{2} \end{aligned}$ to keep the equation balanced.
::第4步:在左侧括号的内部添加此数字。在右侧, 您需要添加 a(b2) 2 来保持方程平衡。

$\begin{aligned} 3 (x^{2} + 3 x + \frac{9}{4}) = 5 + \frac{27}{4} \end{aligned}$

::3(x2+3x+94=5+274)

Step 5: Factor the left side and simplify the right.
::第5步:以左侧为因子,简化右侧。

$\begin{aligned} 3 {(x + \frac{3}{2})}^{2} = \frac{47}{4} \end{aligned}$

::3(x+322)=474

Step 6: Solve by using square roots.
::步骤6:用平方根解决。

$\begin{aligned} {(x + \frac{3}{2})}^{2} & = \frac{47}{12} \\ x + \frac{3}{2} & = \pm \frac{\sqrt{47}}{\sqrt{12}} \\ x & = - \frac{3}{2} \pm \frac{\sqrt{47}}{2 \sqrt{3}} \\ x & = - \frac{3}{2} \pm \frac{\sqrt{141}}{6} \end{aligned}$

:x+32)2=4712x+324712x324723x321416

However, because x is the length of the parallelogram's base, it must have a positive value. Only $\begin{aligned} x = - \frac{3}{2} + \frac{\sqrt{141}}{6} \end{aligned}$ results in a positive value, so the length of the base is $\begin{aligned} x = - \frac{3}{2} + \frac{\sqrt{141}}{6} \end{aligned}$ .
::然而,由于 x 是平行图基数的长度, 它必须有一个正值。只有 x32+1416 得出正值, 所以基数的长度是 x32+1416 。

Example 2
::例2

Solve the following quadratic equation by completing the square: $\begin{aligned} 5 x^{2} + 29 x - 6 = 0 \end{aligned}$ .
::通过完成方形( 5x2+29x-6=0) 解决以下二次方程。

$\begin{aligned} 5 x^{2} + 29 x - 6 & = 0 \\ 5 (x^{2} + \frac{29}{5} x) & = 6 \\ 5 (x^{2} + \frac{29}{5} x + \frac{841}{100}) & = 6 + \frac{841}{20} \\ 5 {(x + \frac{29}{10})}^{2} & = \frac{961}{20} \\ {(x + \frac{29}{10})}^{2} & = \frac{961}{100} \\ x + \frac{29}{10} & = \pm \frac{31}{10} \\ x & = - \frac{29}{10} \pm \frac{31}{10} \\ x & = - 6, \frac{1}{5} \end{aligned}$

::5x2+29x-6=05(x2+295x)=65(x2+295x+841100)=6+841205(x+291010)2=96120(x+2910)2=961100x+2910=961100x+2910=3110x_2910x_2910_3110x_6,15

Example 3
::例3

Solve the following quadratic by completing the square: $\begin{aligned} 8 x^{2} - 32 x + 4 = 0 \end{aligned}$ .
::通过完成正方形( 8x2- 32x+4=0) 解决以下二次方形。

$\begin{aligned} 8 x^{2} - 32 x + 4 & = 0 \\ 8 (x^{2} - 4 x) & = - 4 \\ 8 (x^{2} - 4 x + 4) & = - 4 + 32 \\ 8 (x - 2)^{2} & = 28 \\ (x - 2)^{2} & = \frac{7}{2} \\ x - 2 & = \pm \frac{\sqrt{7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} \\ x & = 2 \pm \frac{\sqrt{14}}{2} \end{aligned}$

::8x2 - 32x+4=08(x2 - 4x)\48(x2 - 4x+4)\}4+328(x-2)2=28(x-2)2=28(x-2)2=72x-272-22x=2142

Review
::回顾

Solve the quadratic equations by completing the square.
::通过完成广场来解决二次方程。

$\begin{aligned} 6 x^{2} - 12 x - 7 = 0 \end{aligned}$
::6x2-12x-7=0

$\begin{aligned} - 4 x^{2} + 24 x - 100 = 0 \end{aligned}$
::-4x2+24x-100=0

$\begin{aligned} 5 x^{2} - 30 x + 55 = 0 \end{aligned}$
::5x2-30x+55=0

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

$\begin{aligned} \frac{1}{2} x^{2} + 7 x + 8 = 0 \end{aligned}$
::12x2+7x+8=0

$\begin{aligned} - 3 x^{2} + 4 x + 15 = 0 \end{aligned}$
::- 3x2+4x+15=0

Solve the following equations by factoring, using square roots, or completing the square.
::通过保理、使用平方根或完成方块来解决以下方程式。

$\begin{aligned} 4 x^{2} - 4 x - 8 = 0 \end{aligned}$
::4x2-4x-8=0

$\begin{aligned} 2 x^{2} + 9 x + 7 = 0 \end{aligned}$
::2x2+9x+7=0

$\begin{aligned} - 5 (x + 4)^{2} - 19 = 26 \end{aligned}$
::-5(x+4)2-19=26

$\begin{aligned} 3 x^{2} + 30 x - 5 = 0 \end{aligned}$
::3x2+30x-5=0

$\begin{aligned} 9 x^{2} - 15 x - 6 = 0 \end{aligned}$
::9x2--15x-6=0

$\begin{aligned} 10 x^{2} + 40 x + 88 = 0 \end{aligned}$
::10x2+40x+88=0

Problems 13-15 build off of each other.
::13-15问题相互交织。

Challenge Complete the square for $\begin{aligned} a x^{2} + b x + c = 0 \end{aligned}$ . Follow the steps outlined in this lesson. Your final answer should be in terms of $\begin{aligned} a, b, \end{aligned}$ and $\begin{aligned} c \end{aligned}$ .
::挑战方块为x2+bx+c=0。遵循本课中概述的步骤。您的最后答案应该是 a、 b 和 c 。

For the equation $\begin{aligned} 8 x^{2} + 6 x - 5 = 0 \end{aligned}$ , use the formula you found in #13 to solve for $\begin{aligned} x \end{aligned}$ .
::方程式 8x2+6x- 5=0 使用在# 13 中找到的公式解析 x。

Is the equation in #14 factorable? If so, factor and solve it.
::#14 中的方程式是可乘的吗? 如果是的话, 系数并解决它。

Error Analysis Examine the worked out problem below.
::检查以下已解决的问题。

$\begin{aligned} 4 x^{2} - 48 x + 11 & = 0 \\ 4 (x^{2} - 12 x + \underline{}) & = - 11 \\ 4 (x^{2} - 12 x + 36) & = - 11 + 36 \\ 4 (x - 6)^{2} & = 25 \\ (x - 6)^{2} & = \frac{25}{4} \\ x - 6 & = \pm \frac{5}{2} \\ x & = 6 \pm \frac{5}{2} \to \frac{17}{2}, \frac{7}{2} \end{aligned}$

::4x2 - 48x+11=04(x2 - 12x) 114(x2 - 12x+36) 11+364(x-6) 2=25(x-6) 2=254x6=254x6*5x=6*52x=652172,72

Plug the answers into the original equation to see if they work. If not, find the error and correct it.
::将答案插入原始方程以查看是否有效。否则, 找到错误并纠正错误。

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

Completing the Square ::完成广场

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)

Completing the Square
::完成广场

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Review
::回顾

Review (Answers)
::回顾(答复)