Course: 5.7 利用平根解决四方

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利用平根解决四方
Mrs. Garber draws a square on the board and writes the equation $\begin{aligned} \frac{4 s^{2}}{5} - 3 = 13 \end{aligned}$ on the board. "This equation represents the area," she says. "What is the length of each side ( s )?"
::Garber夫人在板上画了一个方块,并在板上写了方程式4s25-3=13。“这个方程式代表了这个区域”,她说,“每一边的长度是多少?”

Solving Quadratics Using Square Roots
::利用平根解决四方

Now that you are familiar with square roots, we will use them to . Keep in mind, that square roots cannot be used to solve every type of quadratic. In order to solve a quadratic equation by using square roots, an $\begin{aligned} x - \end{aligned}$ term cannot be present. Solving a quadratic equation by using square roots is very similar to solving a linear equation . In the end, you must isolate the $\begin{aligned} x^{2} \end{aligned}$ or whatever is being squared.
::既然您熟悉正方根, 我们将使用它们。记住, 平方根不能用来解决每一种四方形。为了通过正方根解决四方形方程式, 无法存在 x- term 。使用正方根解决四方形与解决线性方程非常相似。最后, 您必须分离 x2 或正方形中的任何方程式。

Solve the following using square roots.
::使用平方根解决以下问题。

$\begin{aligned} 2 x^{2} - 3 = 15 \end{aligned}$

::2x2-3=15

Start by isolating the $\begin{aligned} x^{2} \end{aligned}$ .
::以分隔 x2 开始。

$\begin{aligned} 2 x^{2} - 3 & = 15 \\ 2 x^{2} & = 18 \\ x^{2} & = 9 \end{aligned}$

::2x2-3=152x2=18x2=9

At this point, you can take the square root of both sides.
::在这一点上,你可以选择双方的平方根。

$\begin{aligned} \sqrt{x^{2}} & = \pm \sqrt{9} \\ x & = \pm 3 \end{aligned}$

::x29x3

Notice that $\begin{aligned} x \end{aligned}$ has two solutions; 3 or -3. When taking the square root, always put the $\begin{aligned} \pm \end{aligned}$ (plus or minus sign) in front of the square root. This indicates that the positive or negative answer will be the solution.
::注意 x 有两个解决方案; 3 或 - 3 。在选择正方根时, 总是在正方根前面放 ( + 或减符号 ) 。这表示正或负答案就是正或负答案。

Check :
::检查 :

$\begin{aligned} 2 (3)^{2} - 3 = 15 2 (- 3)^{2} - 3 = 15 \\ 2 \cdot 9 - 3 = 15 o r 2 \cdot 9 - 3 = 15 \\ 18 - 3 = 15 18 - 3 = 15 \end{aligned}$

::2(3)2-3=152(-3-3)2-3-3=15 2__9-3=15or 2__9-3=15or 2__9-3=3=15-18-3=1518-3=15

Solve the following using square roots.
::使用平方根解决以下问题。

$\begin{aligned} \frac{x^{2}}{16} + 3 = 27 \end{aligned}$

::x216+3=27

Isolate $\begin{aligned} x^{2} \end{aligned}$ and then take the square root.
::孤立 x2 , 然后取平方根。

$\begin{aligned} \frac{x^{2}}{16} + 3 & = 27 \\ \frac{x^{2}}{16} & = 24 \\ x^{2} & = 384 \\ x & = \pm \sqrt{384} = \pm 8 \sqrt{6} \end{aligned}$

::x216+3=27x216=24x2=384x38486

Solve the following using square roots.
::使用平方根解决以下问题。

$\begin{aligned} 3 (x - 5)^{2} + 7 = 43 \end{aligned}$

::3(x--5)2+7=43

In this problem , $\begin{aligned} x \end{aligned}$ is not the only thing that is squared. Isolate the $\begin{aligned} (x - 5)^{2} \end{aligned}$ and then take the square root.
::在此问题上, x 不是唯一的正方。分离( x-5) 2, 然后取平方根。

$\begin{aligned} 3 (x - 5)^{2} + 7 & = 43 \\ 3 (x - 5)^{2} & = 36 \\ (x - 5)^{2} & = 12 \\ x - 5 & = \pm \sqrt{12} o r \pm 2 \sqrt{3} \end{aligned}$

::3(x--5)2+7=433(x-5)2=36(x-5)2=12x-512或23

Now that the square root is gone, add 5 to both sides.
::平方根已经没了双方加五根

$\begin{aligned} x - 5 & = \pm 2 \sqrt{3} \\ x & = 5 \pm 2 \sqrt{3} \end{aligned}$

::x-523x=523

$\begin{aligned} x = 5 + 2 \sqrt{3} \end{aligned}$ or $\begin{aligned} 5 - 2 \sqrt{3} \end{aligned}$ . We can estimate these solutions as decimals; 8.46 or 1.54. Remember, that the most accurate answer includes the radical numbers.
::x=5+23 或 5- 23。我们可以将这些解算方法估计为小数数; 8.46 或 1.54。记住, 最准确的答案包括激进数字。

Examples
::实例

Example 1
::例1

Earlier, you were asked to find the length of each side.
::早些时候,你被要求找到每一边的长度。

To find s , isolate $\begin{aligned} s^{2} \end{aligned}$ and then take the square root.
::要找到 s, 分离 S2 然后取平方根。

$\begin{aligned} \frac{4 s^{2}}{5} - 3 & = 13 \\ \frac{4 s^{2}}{5} & = 16 \\ s^{2} & = 20 \\ s & = \pm \sqrt{20} = \pm 2 \sqrt{5} \end{aligned}$

::4s25 - 3=134s25=16s2=20s=20s2025

Therefore the length of the square's side is $\begin{aligned} 2 \sqrt{5} \end{aligned}$ .
::因此,广场一侧的长度是25。

Example 2
::例2

Solve the following quadratic equation.
::解决以下的二次方程。

$\begin{aligned} \frac{2}{3} x^{2} - 14 = 38 \end{aligned}$
::23x2-14=38

Isolate $\begin{aligned} x^{2} \end{aligned}$ and take the square root.
::孤立 x2 并取平方根。

$\begin{aligned} \frac{2}{3} x^{2} - 14 & = 38 \\ \frac{2}{3} x^{2} & = 52 \\ x^{2} & = 78 \\ x & = \pm \sqrt{78} \end{aligned}$

::23x2-14=3823x2=52x2=78x78

Example 3
::例3

Solve the following quadratic equation.
::解决以下的二次方程。

$\begin{aligned} 11 + x^{2} = 4 x^{2} + 5 \end{aligned}$
::11+x2=4x2+5

Combine all like terms , then isolate $\begin{aligned} x^{2} \end{aligned}$ .
::组合所有类似条件, 然后分离 x2 。

$\begin{aligned} 11 + x^{2} & = 4 x^{2} + 5 \\ - 3 x^{2} & = - 6 \\ x^{2} & = 2 \\ x & = \pm \sqrt{2} \end{aligned}$

::11+x2=4x2+5-3x2=6x2=2x#2

Example 4
::例4

Solve the following quadratic equation.
::解决以下的二次方程。

$\begin{aligned} (2 x + 1)^{2} - 6 = 19 \end{aligned}$
:2x+1)2-6=19

Isolate what is being squared, take the square root, and then isolate $\begin{aligned} x \end{aligned}$ .
::分离正方形,取平方根,然后分离 x。

$\begin{aligned} (2 x + 1)^{2} - 6 & = 19 \\ (2 x + 1)^{2} & = 25 \\ 2 x + 1 & = \pm 5 \\ 2 x & = - 1 \pm 5 \\ x & = \frac{- 1 \pm 5}{2} \to x = \frac{- 1 + 5}{2} = 2 o r x = \frac{- 1 - 5}{2} = - 3 \end{aligned}$

:2x+1)2 -6=19(2x+1)2=252x+1=252x+1*************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************************

Review
::回顾

Solve the following quadratic equations. Reduce answers as much as possible. No decimals.
::解决以下的二次方程。尽可能减少答案, 不小数。

$\begin{aligned} x^{2} = 144 \end{aligned}$
::x2=144x2=144

$\begin{aligned} 5 x^{2} - 4 = 16 \end{aligned}$
::5x2-4=16=16

$\begin{aligned} 8 - 10 x^{2} = - 22 \end{aligned}$
::8-10x222

$\begin{aligned} (x + 2)^{2} = 49 \end{aligned}$
:x+2)2=49

$\begin{aligned} 6 (x - 5)^{2} + 1 = 19 \end{aligned}$
::6(xx-5-5)2+1=19

$\begin{aligned} \frac{3}{4} x^{2} - 19 = 26 \end{aligned}$
::34x2-19=26

$\begin{aligned} x^{2} - 12 = 36 - 2 x^{2} \end{aligned}$
::x2 - 12=36 - 2x2

$\begin{aligned} 9 - \frac{x^{2}}{3} = - 33 \end{aligned}$
::9 - x23 33

$\begin{aligned} - 4 (x + 7)^{2} = - 52 \end{aligned}$
::- 4(x+7)252

$\begin{aligned} 2 (3 x + 4)^{2} - 5 = 45 \end{aligned}$
::2(3x+4)2-5=45

$\begin{aligned} \frac{1}{3} (x - 10)^{2} - 8 = 16 \end{aligned}$
::13(x-10)2-8=16

$\begin{aligned} \frac{(x - 1)^{2}}{6} - \frac{8}{3} = \frac{7}{2} \end{aligned}$
:x-1)26-83=72

Use either factoring or solving by square roots to solve the following quadratic equations.
::使用保理或平方根解决以下四方形方程式。

$\begin{aligned} x^{2} - 16 x + 55 = 0 \end{aligned}$
::x2 - 16x+55=0

$\begin{aligned} 2 x^{2} - 9 = 27 \end{aligned}$
::2x2-9=27

$\begin{aligned} 6 x^{2} + 23 x = - 20 \end{aligned}$
::6x2+23xx @ @ @% @%20

Writing Write a set of hints that will help you remember when you should solve an equation by factoring and by square roots. Are there any quadratics that can be solved using either method?
::写入一组提示, 帮助您记住您何时应该通过乘数和正方根解析方程式。是否有任何二次方程式可以使用这两种方法解析 ?

Solve $\begin{aligned} x^{2} - 9 = 0 \end{aligned}$ by factoring and by using square roots. Which do you think is easier? Why?
::通过乘数和用正方根解决 x2- 9=0。您认为哪一种比较容易? 为什么?

Solve $\begin{aligned} (3 x - 2)^{2} + 1 = 17 \end{aligned}$ by using square roots. Then, solve $\begin{aligned} 3 x^{2} - 4 x - 4 = 0 \end{aligned}$ by factoring. What do you notice? What can you conclude?
::使用平方根解决( 3x-2-2) 2+1=17 。然后通过乘数解决 3x2- 4x-4=0 。您注意到什么 ? 您可以得出什么结论 ?

Real Life Application The aspect ratio of a TV screen is the ratio of the screen’s width to its height. For HDTVs, the aspect ratio is 16:9. What is the width and height of a 42 inch screen TV? (42 inches refers to the length of the screen’s diagonal.) HINT: Use the Pythagorean Theorem. Round your answers to the nearest hundredth.
::电视屏幕的侧边比是屏幕宽度与高度之比。对于HDTV来说,侧边比是 16:9. 42 英寸屏幕电视的宽度和高度是 16:9. (42 英寸是指屏幕对角的长度 ) 。 HINT:使用 Pythagoren 理论。将您的答案转至最接近的百分百位。

Real Life Application When an object is dropped, its speed continually increases until it reaches the ground. This scenario can be modeled by the equation $\begin{aligned} h = - 16 t^{2} + h_{0} \end{aligned}$ , where $\begin{aligned} h \end{aligned}$ is the height, $\begin{aligned} t \end{aligned}$ is the time (in seconds), and $\begin{aligned} h_{0} \end{aligned}$ is the initial height of the object. Round your answers to the nearest hundredth.

If you drop a ball from 200 feet, what is the height after 2 seconds?
::如果你从200英尺投下一个球两秒后高度是多少?

After how many seconds will the ball hit the ground?
::几秒后球会打到地上?

::当一个对象被丢弃时, 其速度会持续递增, 直到它到达地面。这个假想可以以 qual h16t2+h0 模式模拟, 该方程式的高度是 h, t 是该对象的初始高度( 秒) 。 h0 是该对象的初始高度。将您的答复绕到最接近的百度。如果您从200 英尺投下一个球, 两秒之后的高度是多少? 在球撞击地面几秒之后?

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

利用平根解决四方

Solving Quadratics Using Square Roots ::利用平根解决四方

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Example 4 ::例4

Review ::回顾

Review (Answers) ::回顾(答复)

Solving Quadratics Using Square Roots
::利用平根解决四方

Examples
::实例

Example 1
::例1

Example 2
::例2

Example 3
::例3

Example 4
::例4

Review
::回顾

Review (Answers)
::回顾(答复)