Course: 10.6 完成广场 | The Way To Learn

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完成广场
Completing the Square
::完成广场

You saw in the last section that if you have a quadratic equation of the form $\begin{aligned} (x - 2)^{2} = 5 \end{aligned}$ , you can easily solve it by taking the square root of each side:
::您在最后一节中看到,如果有表( x-2) 2=5 的二次方程,您可以很容易地以每面的平方根来解答它:

$\begin{aligned} x - 2 = \sqrt{5} and x - 2 = - \sqrt{5} \end{aligned}$

::x-2=5和 x-2=5 和 x-2=5

Simplify to get:
::要获得的简化 :

$\begin{aligned} x = 2 + \sqrt{5} \approx 4.24 and x = 2 - \sqrt{5} \approx - 0.24 \end{aligned}$

::x=2+54.24和 x=2-50.24

So what do you do with an equation that isn’t written in this nice form? In this section, you’ll learn how to rewrite any quadratic equation in this form by .
::那么,你对不是以这样好的形式写成的方程式有什么用呢?在这一节中,你将学会如何用这种形式重写任何二次方程式。

Complete the Square of a Quadratic Expression
::完成二次曲线表达式的广场

Completing the square lets you rewrite a quadratic expression so that it contains a perfect square trinomial that you can factor as the square of a binomial .
::完成方形后, 您可以重写一个二次表达式, 这样它就包含一个完美的正方方形三角, 您可以将它作为二进制的正方形。

Remember that the square of a binomial takes one of the following forms:
::记住二进制的正方形具有以下一种形式:

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

:x+a)2=x2+2x2+2ax+a2(x-a)2=x2-2ax+a2

So in order to have a perfect square trinomial, we need two terms that are perfect squares and one term that is twice the product of the square roots of the other terms.
::因此,为了有一个完全的平方方形三角体, 我们需要两个完美的平方体, 一个是其他词的平方根的产物的两倍。

Completing the Square
::完成广场

1. Complete the square for the quadratic expression $\begin{aligned} x^{2} + 4 x \end{aligned}$ .
::1. 完成四边表达式x2+4x的方形。

To complete the square we need a constant term that turns the expression into a perfect square trinomial. Since the middle term in a perfect square trinomial is always 2 times the product of the square roots of the other two terms, we re-write our expression as:
::要完成方形,我们需要一个常数术语,将表达方式转换成完美的平方三角。由于在完全平方三角中,中间任期总是另外两个词的平方根产物的2倍,因此我们将表达方式重写为:

$\begin{aligned} x^{2} + 2 (2) (x) \end{aligned}$

::x2+2(2)(x)

We see that the constant we are seeking must be $\begin{aligned} 2^{2} : \end{aligned}$
::我们认为,我们所寻求的常数必须是22:

$\begin{aligned} x^{2} + 2 (2) (x) + 2^{2} \end{aligned}$

::x2+2(2)(x)+22

Answer: By adding 4 to both sides, this can be factored as: $\begin{aligned} (x + 2)^{2} \end{aligned}$
::答复:双方增加4个,可以作为(x+2)2的考虑因素。

Notice, though, that we just changed the value of the whole expression by adding 4 to it. If it had been an equation, we would have needed to add 4 to the other side as well to make up for this.
::但请注意,我们刚刚改变了整个表达方式的值,增加了4个。如果这是一个方程的话,我们需要在另一边加上4个,并弥补这一点。

Also, this was a relatively easy example because $\begin{aligned} a \end{aligned}$ , the coefficient of the $\begin{aligned} x^{2} \end{aligned}$ term, was 1. When that coefficient doesn’t equal 1, we have to factor it out from the whole expression before completing the square.
::而且,这是一个比较容易的例子,因为一个X2术语的系数是1,当该系数不等于1时,我们必须在完成正方形之前从整个表达式中将其考虑在内。

2. Complete the square for the quadratic expression $\begin{aligned} 4 x^{2} + 32 x \end{aligned}$ .
::2. 4x2+32x四方形表达式完成方形。

Factor the coefficient of the $\begin{aligned} x^{2} \end{aligned}$ term:
::x2 术语的系数系数 :

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

Re-write the expression:
::重写表达式 :

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

We complete the square by adding the constant $\begin{aligned} 4^{2} \end{aligned}$ :
::我们通过添加常数42来完成广场:

$\begin{aligned} 4 (x^{2} + 2 (4) (x) + 4^{2}) \end{aligned}$
::4(x2+2(4)(x)+42)

Factor the perfect square trinomial inside the parenthesis:
::括号内完美的正方形三角数乘以 :

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

The expression “completing the square” comes from a geometric interpretation of this situation. Let’s revisit the quadratic expression in Example 1: $\begin{aligned} x^{2} + 4 x \end{aligned}$ .
::“完成广场”一词来自对这种情况的几何解释。让我们重温例1: x2+4x中的二次表达式。

We can think of this expression as the sum of three areas. The first term represents the area of a square of side $\begin{aligned} x \end{aligned}$ . The second expression represents the areas of two rectangles with a length of 2 and a width of $\begin{aligned} x \end{aligned}$ :
::我们可以将这个表达式视为三个区域的总和。第一个术语代表侧面 x 的平方区域。第二个表达式代表两个矩形的区域,其长度为 2 和 x 的宽度为 x :

We can combine these shapes as follows:
::我们可以将这些形状合并如下:

We obtain a square that is not quite complete. To complete the square, we need to add a smaller square of side length 2.
::我们得到的正方形不完全,要完成正方形,我们需要增加一个小的正方形的侧长2。

We end up with a square of side length $\begin{aligned} (x + 2) \end{aligned}$ ; its area is therefore $\begin{aligned} (x + 2)^{2} \end{aligned}$ . Let’s demonstrate the method of completing the square with an example.
::我们最终会有一个侧长的平方(x+2);因此,其面积是(x+2)2。让我们举一个例子来展示完成平方的方法。

Solving for Unknown Values
::解决未知值

Solve the following quadratic equation: $\begin{aligned} 3 x^{2} - 10 x = - 1 \end{aligned}$
::解决以下二次方程式: 3x2- 10x1

Divide all terms by the coefficient of the $\begin{aligned} x^{2} \end{aligned}$ term:
::将所有条件除以 x2 术语的系数 :

$\begin{aligned} x^{2} - \frac{10}{3} x = - \frac{1}{3} \end{aligned}$

::x2 - 103x% 13

Rewrite: $\begin{aligned} x^{2} - 2 (\frac{5}{3}) (x) = - \frac{1}{3} \end{aligned}$
::重写: x2-2 (53) (x) {______________________________________________________________________________________________________________

In order to have a perfect square trinomial on the right-hand-side we need to add the constant $\begin{aligned} {(\frac{5}{3})}^{2} \end{aligned}$ . Add this constant to both sides of the equation:
::为了在右侧有一个完美的正方形三角形,我们需要加上常数(53)2。在等式的两侧加上这个常数:

$\begin{aligned} x^{2} - 2 (\frac{5}{3}) (x) + {(\frac{5}{3})}^{2} = - \frac{1}{3} + {(\frac{5}{3})}^{2} \end{aligned}$

::x2-2( 53)(x) +( 532) 13+( 533) 2

Factor the perfect square trinomial and simplify:
::将完美的平方三角和简化乘以:

$\begin{aligned} {(x - \frac{5}{3})}^{2} & = - \frac{1}{3} + \frac{25}{9} \\ {(x - \frac{5}{3})}^{2} & = \frac{22}{9} \end{aligned}$

:x- 532=229)13+259(x-53)2=229

Take the square root of both sides:
::以双方的平方根:

$\begin{aligned} x - \frac{5}{3} & = \sqrt{\frac{22}{9}} & and & x - \frac{5}{3} = - \sqrt{\frac{22}{9}} \\ x & = \frac{5}{3} + \sqrt{\frac{22}{9}} \approx 3.23 & and & x = \frac{5}{3} - \sqrt{\frac{22}{9}} \approx 0.1 \end{aligned}$

::x-53=229和x-53=229和x-53=229*229x53+229*3.23和x=53-229*0.1

Answer: $\begin{aligned} x = 3.23 \end{aligned}$ and $\begin{aligned} x = 0.1 \end{aligned}$
::答复:x=3.23和x=0.1

Solving Quadratic Equations in Standard Form
::标准表格中溶解二次赤道

If an equation is in standard form $\begin{aligned} (a x^{2} + b x + c = 0) \end{aligned}$ , we can still solve it by the method of completing the square. All we have to do is start by moving the constant term to the right-hand-side of the equation.
::如果方程式以标准形式( 轴+bx+c=0) , 我们仍可以通过完成方块的方法解答它。我们只需要从常数开始, 将常数移到方程式的右侧。

Solve the following quadratic equation: $\begin{aligned} x^{2} + 15 x + 12 = 0 \end{aligned}$
::解决以下二次方程式: x2+15x+12=0

Move the constant to the other side of the equation:
::将常数移动到方程式的另一侧:

$\begin{aligned} x^{2} + 15 x = - 12 \end{aligned}$

::x2+15x% 12

Rewrite: $\begin{aligned} x^{2} + 2 (\frac{15}{2}) (x) = - 12 \end{aligned}$
::重写 : x2+2( 152)( x) @%% 12

Add the constant $\begin{aligned} {(\frac{15}{2})}^{2} \end{aligned}$ to both sides of the equation:
::在方程的两侧添加常数( 152) 2:

$\begin{aligned} x^{2} + 2 (\frac{15}{2}) (x) + {(\frac{15}{2})}^{2} = - 12 + {(\frac{15}{2})}^{2} \end{aligned}$

::x2+2( 152)(x) +( 152) +( 152) +( 12)+( 152) 2

Factor the perfect square trinomial and simplify:
::将完美的平方三角和简化乘以:

$\begin{aligned} {(x + \frac{15}{2})}^{2} & = - 12 + \frac{225}{4} \\ {(x + \frac{15}{2})}^{2} & = \frac{177}{4} \end{aligned}$

:x+152)212+2254(x+152)2=1774

Take the square root of both sides:
::以双方的平方根:

$\begin{aligned} x + \frac{15}{2} & = \sqrt{\frac{177}{4}} & and & x + \frac{15}{2} = - \sqrt{\frac{177}{4}} \\ x & = - \frac{15}{2} + \sqrt{\frac{177}{4}} \approx - 0.85 & and & x = - \frac{15}{2} - \sqrt{\frac{177}{4}} \approx - 14.15 \end{aligned}$

::x+152=1774和x+15217474x}152+174474}0.85和x}152 -17447}14.15

Answer: $\begin{aligned} x = - 0.85 \end{aligned}$ and $\begin{aligned} x = - 14.15 \end{aligned}$
::答复:x0.85和x14.15

Example
::示例示例示例示例

Example 1
::例1

Solve the following quadratic equation: $\begin{aligned} - x^{2} + 22 x = 5 \end{aligned}$
::解决以下二次方程:-x2+22x=5

Divide all terms by the coefficient of the $\begin{aligned} x^{2} \end{aligned}$ term:
::将所有条件除以 x2 术语的系数 :

$\begin{aligned} x^{2} - 22 x = - 6 \end{aligned}$

::x2 - 22x% 6

Rewrite: $\begin{aligned} x^{2} - 2 (11) (x) = - 6. \end{aligned}$
::重写: x2-2( 11) (x) @% 6 。

In order to have a perfect square trinomial on the right-hand-side we need to add the constant $\begin{aligned} {(11)}^{2} \end{aligned}$ . Add this constant to both sides of the equation:
::为了在右侧有一个完美的正方形三角形,我们需要加上常数(112)。在方程的两侧加上这个常数:

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

::x2-2(11)(x)+(11)2+6+(11)2

Factor the perfect square trinomial and simplify:
::将完美的平方三角和简化乘以:

$\begin{aligned} {(x - 11)}^{2} & = - 6 + (11)^{2} \\ {(x - \frac{5}{3})}^{2} & = 16 \end{aligned}$

:x-112)6+(112)2(x-53)2=16

Take the square root of both sides:
::以双方的平方根:

$\begin{aligned} x - 11 & = \sqrt{16} & and & x - 11 = - \sqrt{16} \\ x & = 11 + \sqrt{16} = 15 & and & x = 11 - \sqrt{4} = 7 \end{aligned}$

::x- 11=16andx- 11_ 11_ 16x=11+16=15andx=11- 4=7

Answer: $\begin{aligned} x = 15 \end{aligned}$ and $\begin{aligned} x = 7 \end{aligned}$
::答复:x=15和x=7

Review
::回顾

Complete the square for each expression.
::填写每个表达式的正方形。

$\begin{aligned} x^{2} + 5 x \end{aligned}$
::x2+5xx

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

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

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

$\begin{aligned} 3 x^{2} + 18 x \end{aligned}$
::3x2+18x

$\begin{aligned} 2 x^{2} - 22 x \end{aligned}$
::2x2-22x

$\begin{aligned} 8 x^{2} - 10 x \end{aligned}$
::8x2 - 10x

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

Solve each quadratic equation by completing the square.
::通过完成广场来解决每个二次方程。

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

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

$\begin{aligned} x^{2} + 10 x + 15 = 0 \end{aligned}$
::x2+10x+15=0

$\begin{aligned} x^{2} + 15 x + 20 = 0 \end{aligned}$
::x2+15x+20=0

$\begin{aligned} 2 x^{2} - 18 x = 0 \end{aligned}$
::2x2 - 18x=0

$\begin{aligned} 4 x^{2} + 5 x = - 1 \end{aligned}$
::4x2+5x%1

$\begin{aligned} 10 x^{2} - 30 x - 8 = 0 \end{aligned}$
::10x2-30x-8=0

$\begin{aligned} 5 x^{2} + 15 x - 40 = 0 \end{aligned}$
::5x2+15x-40=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。

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