Course: 11.6 激进等量 | The Way To Learn

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激进等量
Radical Equations
::激进等量

When the variable in an equation appears inside a radical sign, the equation is called a radical equation. To solve a radical equation, we need to eliminate the radical and change the equation into a polynomial equation.
::当一个方程式中的变量出现在一个激进符号中时,该方程式就被称为激进方程式。要解决激进方程式,我们需要消除激进方程式,并将方程式转换成多元方程式。

A common method for solving radical equations is to isolate the most complicated radical on one side of the equation and raise both sides of the equation to the power that will eliminate the radical sign. If there are any radicals left in the equation after simplifying, we can repeat this procedure until all radical signs are gone. Once the equation is changed into a polynomial equation, we can solve it with the methods we already know.
::解决激进方程式的一个常见方法是将方程式一边最复杂的激进分子隔离开来,并将方程式的两边推向消除激进标志的力量。如果在简化后方程式中还剩下任何激进分子,我们可以重复这一程序,直到所有激进迹象消失为止。一旦方程式变成一个多元方程式,我们就可以用我们已经知道的方法来解决。

We must be careful when we use this method, because whenever we raise an equation to a power, we could introduce false solutions that are not in fact solutions to the original problem. These are called extraneous solutions. In order to make sure we get the correct solutions, we must always check all solutions in the original radical equation.
::我们必须在使用这种方法时小心谨慎,因为每当我们向力量提出等式时,我们可以引入虚假的解决办法,这些办法实际上不是解决最初问题的办法,这些办法被称为不相干的解决办法。为了确保我们得到正确的解决办法,我们必须始终在原始激进的等式中检查所有解决办法。

Solve a Radical Equation
::解决激进方程式

Let’s consider a few simple examples of radical equations where only one radical appears in the equation.
::让我们考虑几个激进方程式的简单例子, 在公式中只出现一个激进方程式。

Finding the Real Solutions of a Radical Equation
::寻找激进方程式的真正解决方案

1. Find the real solutions of the equation $\begin{align*}\sqrt{2x-1}=5\end{align*}$ .
::1. 找出2x-1=5等式的真正解决办法。

Since the radical expression is already isolated, we can just square both sides of the equation in order to eliminate the radical sign:
::因为激进的表达已经是孤立的,

$\begin{aligned} {(\sqrt{2 x - 1})}^{2} = 5^{2} \end{aligned}$
$\begin{align*}\left(\sqrt{2x-1}\right)^2=5^2\end{align*}$
:2x-1)2=52

$\begin{aligned} Remember that \sqrt{a^{2}} = a so the equation simplifies to: & 2 x - 1 & = 25 \\ Add one to both sides: & 2 x & = 26 \\ Divide both sides by 2: & \underline{\underline{x = 13}} \end{aligned}$
$\begin{align*}\text{Remember that} \ \sqrt{a^2}=a \ \text{so the equation simplifies to:} && 2x-1& =25\\ \text{Add one to both sides:} && 2x& =26\\ \text{Divide both sides by 2:} &&& \underline{\underline{x=13}}\end{align*}$
::记住 a2=a 所以方程式简化为: 2x- 1=25 向两侧都添加一个 : 2x=26 以 2: x=13 将两侧的方程式简化为 2: x=13 。

Finally we need to plug the solution in the original equation to see if it is a valid solution.
::最后,我们需要在最初的方程式中插入解决方案,看看它是否有效。

$\begin{align*}\sqrt{2x-1}=\sqrt{2(13)-1}=\sqrt{26-1}=\sqrt{25}=5\end{align*}$ The solution checks out.
::2x-1=2(13)-1=26-1=1=25=5 验证解决方案。

2. Find the real solutions of $\begin{align*}\sqrt[3]{3-7x}-3=0\end{align*}$ .
::2. 找到3-7x3-3-3=0的真正解决办法。

$\begin{aligned} We isolate the radical on one side of the equation: & \sqrt[3]{3 - 7 x} & = 3 \\ Raise each side of the equation to the third power: & {(\sqrt[3]{3 - 7 x})}^{3} & = 3^{3} \\ Simplify: & 3 - 7 x & = 27 \\ Subtract 3 from each side: & - 7 x & = 24 \\ Divide both sides by -7: & \underline{\underline{x = - \frac{24}{7}}} \end{aligned}$
$\begin{align*}\text{We isolate the radical on one side of the equation:} && \sqrt[3]{3-7x}& =3\\ \text{Raise each side of the equation to the third power:} && \left(\sqrt[3]{3-7x}\right)^3& =3^3\\ \text{Simplify:} && 3-7x& =27\\ \text{Subtract 3 from each side:} && -7x& =24\\ \text{Divide both sides by -7:} &&& \underline{\underline{x=-\frac{24}{7}}}\end{align*}$
::我们从方程的一面分离基数 : 3 - 7x3= 3 将方程的两侧排出到第三列 : (3 - 7x3) 3 = 33 简化: 3 - 7x= 27 分数 3 : - 7x= 24Divide 双方由 - 7: x\\ 247 =

Check: $\begin{align*}\sqrt[3]{3-7x}-3=\sqrt[3]{3-7 \left(-\frac{24}{7}\right)}-3=\sqrt[3]{3+24}-3=\sqrt[3]{27}-3=3-3=0\end{align*}$ . The solution checks out.
::检查:3-7x3-3=3-7(-2473-3=3+243-3=273-3=3-3=0)。

3. Find the real solutions of $\begin{align*}\sqrt{10-x^2}-x=2\end{align*}$ .
::3. 找到10-x2-x=2的真正解决办法。

$\begin{aligned} We isolate the radical on one side of the equation: & \sqrt{10 - x^{2}} & = 2 + x \\ Square each side of the equation: & {(\sqrt{10 - x^{2}})}^{2} & = (2 + x)^{2} \\ Simplify: & 10 - x^{2} & = 4 + 4 x + x^{2} \\ Move all terms to one side of the equation: & 0 & = 2 x^{2} + 4 x - 6 \\ Solve using the quadratic formula: & x & = \frac{- 4 \pm \sqrt{4^{2} - 4 (2) (- 6)}}{4} \\ Simplify: & x & = \frac{- 4 \pm \sqrt{64}}{4} \\ Re-write \sqrt{24} in simplest form: & x & = \frac{- 4 \pm 8}{4} \\ Reduce all terms by a factor of 2: & x & = 1 or x = - 3 \end{aligned}$
$\begin{align*}\text{We isolate the radical on one side of the equation:} && \sqrt{10-x^2}& =2+x\\ \text{Square each side of the equation:} && \left(\sqrt{10-x^2}\right)^2& =(2+x)^2\\ \text{Simplify:} && 10-x^2& =4+4x+x^2\\ \text{Move all terms to one side of the equation:} && 0& =2x^2+4x-6\\ \text{Solve using the quadratic formula:} && x& =\frac{-4 \pm \sqrt{4^2-4(2)(-6)}}{4}\\ \text{Simplify:} && x& =\frac{-4 \pm \sqrt{64}}{4}\\ \text{Re-write} \ \sqrt{24} \ \text{in simplest form:} && x& =\frac{-4 \pm 8}{4}\\ \text{Reduce all terms by a factor of 2:} && x& =1 \ \text{or} \ x=-3\end{align*}$
::我们将方程的一面: 10- x2=2+xSquare 方程的一面的基数分隔开来: 10- x2=2+x2 2=( 2+x) 2 简化: 10- x2=4+4x+4x+x2 +x2 将所有条件切换到方程的一面: 0=2x2+4x- 4x-6Solve, 使用二次方形公式: x4- 42-4-4-4(2)- 6) 4Square: x4- 644 re- write 24, 以最简单的形式: x=4- 484Remedu 全部条件均以 2:x=1 或 x%3 的乘数进行简化: x=4- 484- 184

Check: $\begin{align*}\sqrt{10-1^2}-1=\sqrt{9}-1=3-1=2\end{align*}$ This solution checks out.
::检查: 10 - 12 - 1=9 - 1=3 - 1=2 此解决方案已检查。

$\begin{align*}\sqrt{10-(-3)^2}-(-3)=\sqrt{1}+3=1+3=4\end{align*}$ This solution does not check out.
::10-(-3-3)2-(-3)=1+3=1+3=1+3=4 此解决方案不退出。

The equation has only one solution, $\begin{align*}\underline{\underline{x=1}}\end{align*}$ ; the solution $\begin{align*}x=-3\end{align*}$ is extraneous.
::方程式只有一个溶液, x=1 ; 溶液 x\ 3 是不相容的。

Applications using Radical Equations
::使用激进等量的应用

Real-World Application: Volume of Spheres
::真实世界应用:球体量

A sphere has a volume of $\begin{align*}456 \ cm^3\end{align*}$ . If the radius of the sphere is increased by 2 cm, what is the new volume of the sphere?
::球体的体积为456厘米3. 如果球体半径增加了2厘米, 球体的新体积是多少?

Make a sketch:
::绘制一张草图:

Define variables: Let $\begin{align*}R =\end{align*}$ the radius of the sphere.
::定义变量:让 R = 球的半径。

Find an equation: The volume of a sphere is given by the formula $\begin{align*}V=\frac{4}{3}\pi R^3\end{align*}$ .
::查找方程式: 公式V=43QR3给出了球体的体积。

Solve the equation:
::解决方程式:

$\begin{aligned} Plug in the value of the volume: & 456 & = \frac{4}{3} π R^{3} \\ Multiply by 3: & 1368 & = 4 π R^{3} \\ Divide by 4 π : & 108.92 & = R^{3} \\ Take the cube root of each side: & R & = \sqrt[3]{108.92} \Rightarrow R = 4.776 c m \\ The new radius is 2 centimeters more: & R & = 6.776 c m \\ The new volume is: & V & = \frac{4}{3} π (6.776)^{3} = \underline{\underline{1302.5}} c m^{3} \end{aligned}$
$\begin{align*}\text{Plug in the value of the volume:} && 456& =\frac{4}{3} \pi R^3\\ \text{Multiply by 3:} && 1368& =4 \pi R^3\\ \text{Divide by} \ 4 \pi: && 108.92& =R^3\\ \text{Take the cube root of each side:} && R& =\sqrt[3]{108.92} \Rightarrow R=4.776 \ cm\\ \text{The new radius is 2 centimeters more:} && R& =6.776 \ cm\\ \text{The new volume is:} && V & =\frac{4}{3} \pi (6.776)^3=\underline{\underline{1302.5}} \ cm^3\end{align*}$
::卷号 : 456= 43QR3 的插件 3: 1368= 43QR3Divide 4: 108.92= R3 选择每侧的立方根 : R= 108.923= 4. 77766 厘米新的半径为 2 厘米以上 : R= 6. 776 厘米新的卷号是: V= 43Q( 6. 776)3= 1302.5 cm3

Check: Let’s plug in the values of the radius into the volume formula:
::选中: 让我们在音量公式中插入半径值 :

$\begin{align*}V=\frac{4}{3} \pi R^3=\frac{4}{3} \pi (4.776)^3=456 \ cm^3\end{align*}$ . The solution checks out.
::V=43QR3=43Q(4.7763=456立方厘米)。

Real-World Application: Kinetic Energy
::现实世界应用:动能

The kinetic energy of an object of mass $\begin{align*}m\end{align*}$ and velocity $\begin{align*}v\end{align*}$ is given by the formula: $\begin{align*}KE=\frac{1}{2} mv^2\end{align*}$ . A baseball has a mass of 145 kg and its kinetic energy is measured to be 654 Joules $\begin{align*}(kg \cdot m^2/s^2)\end{align*}$ when it hits the catcher’s glove. What is the velocity of the ball when it hits the catcher’s glove?
::KE=12mv2. 棒球的重量为145公斤,其动能测量为654 Joules(kgm2/s2),当它击中捕手的手套时。当球击中捕手的手套时,球速是多少?

$\begin{aligned} Start with the formula: & K E & = \frac{1}{2} m v^{2} \\ Plug in the values for the mass and the kinetic energy: & 654 \frac{k g \cdot m^{2}}{s^{2}} & = \frac{1}{2} (145 k g) v^{2} \\ Multiply both sides by 2: & 1308 \frac{k g \cdot m^{2}}{s^{2}} & = 145 k g \cdot v^{2} \\ Divide both sides by 145 k g : & 9.02 \frac{m^{2}}{s^{2}} & = v^{2} \\ Take the square root of both sides: & v & = \sqrt{9.02} \sqrt{\frac{m^{2}}{s^{2}}} = 3.003 m / s \end{aligned}$
$\begin{align*}\text{Start with the formula:} && KE& =\frac{1}{2} mv^2\\ \text{Plug in the values for the mass and the kinetic energy:} && 654 \frac{kg \cdot m^2}{s^2}& =\frac{1}{2}(145\ kg)v^2\\ \text{Multiply both sides by 2:} && 1308 \frac{kg \cdot m^2}{s^2}& =145 \ kg \cdot v^2\\ \text{Divide both sides by 145} \ kg: && 9.02 \frac{m^2}{s^2}& =v^2\\ \text{Take the square root of both sides:} && v& =\sqrt{9.02} \sqrt{\frac{m^2}{s^2}}=3.003 \ m/s\end{align*}$
::以公式开始: KE= 12mv2Plug, 质量值和动能值为: 654kgm2s2= 12(145公斤)v2, 双侧2 减2 1308kgm2s2 = 145kgv2Divide 双侧145k: 9.02m2s2s2= v2, 横跨两侧的平方根:v= 9.02m2s2= 3. 003 ms/s

Check: Plug the values for the mass and the velocity into the energy formula:
::选中 : 将质量值和速度值插入能量公式 :

$\begin{aligned} K E = \frac{1}{2} m v^{2} = \frac{1}{2} (145 k g) (3.003 m / s)^{2} = 654 k g \cdot m^{2} / s^{2} \end{aligned}$
$\begin{align*}KE=\frac{1}{2}mv^2=\frac{1}{2}(145 \ kg)(3.003 \ m/s)^2=654 \ kg \cdot m^2/s^2\end{align*}$
::KE=12mv2=12(145公斤)(3.003米/秒)2=654千克/平方米/秒)2

Example
::示例示例示例示例

Example 1
::例1

Find the real solutions of $\begin{align*}\sqrt{3x-9}-1=5\end{align*}$ .
::找到 3x- 9- 1=5 的真正解决方案。

$\begin{aligned} We isolate the radical on one side of the equation: & \sqrt{3 x - 9} & = 6 \\ Square each side of the equation: & {\sqrt{3 x - 9}}^{2} & = 6^{2} \\ Simplify: & 3 x - 9 & = 36 \\ Add 9 from each side: & 3 x & = 45 \\ Divide both sides by 3: & \underline{\underline{x = \frac{45}{3} = 15}} \end{aligned}$
$\begin{align*}\text{We isolate the radical on one side of the equation:} && \sqrt{3x-9}&=6\\ \text{Square each side of the equation:} && \sqrt{3x-9}^2&=6^2\\ \text{Simplify:} && 3x-9&=36\\ \text{Add 9 from each side:} && 3x&=45\\ \text{Divide both sides by 3:} &&& \underline{\underline{x=\frac{45}{3}=15}}\end{align*}$
::我们分离方程的一面: 3x- 9=6 方程的两侧: 3x- 92=62 方程的两侧简化: 3x- 9=36 Add 9 每侧: 3x=45 Divide 双方在 3: x=453=15 时分立 : x=453=15

Check: $\begin{align*}\sqrt{3x-9}-1=\sqrt{3(15)-9}-1=\sqrt{45-9}-1=\sqrt{36}-1=6-1=5.\end{align*}$ The solution checks out.
::检查:3x- 9- 1=3(15)- 9- 1=45- 9- 1=36- 1=6- 1=5。

Review
::回顾

Find the solution to each of the following radical equations.
::找到解决以下每个极端方程式的方法。

$\begin{align*}\sqrt{x+2}-2=0\end{align*}$
::x+2-2=0

$\begin{align*}\sqrt{3x-1}=5\end{align*}$
::3x-1=5

$\begin{align*}2 \sqrt{4-3x}+3=0\end{align*}$
::24-3x+3=0

$\begin{align*}\sqrt[3]{x-3}=1\end{align*}$
::x-3=1

$\begin{align*}\sqrt[4]{x^2-9}=2\end{align*}$
::x2 - 94=2

$\begin{align*}\sqrt[3]{-2-5x}+3=0\end{align*}$
::--2-5x3+3=0

$\begin{align*}\sqrt{x^2-3}=x-1\end{align*}$
::x2-3=x-1

$\begin{align*}\sqrt{x}=x-6\end{align*}$
::x=x- 6x=x- 6

$\begin{align*}\sqrt{x^2-5x}-6=0\end{align*}$
::x2 - 5x-6=0

$\begin{align*}\sqrt{(x+1)(x-3)}=x\end{align*}$
:x+1)(x-3)=x

$\begin{align*}\sqrt{x+6}=x+4\end{align*}$
::x+6=x+4 x6=x+4

$\begin{align*}\sqrt{3x+4}=-6\end{align*}$
::3x+46

The area of a triangle is $\begin{align*}24 \ in^2\end{align*}$ and the height of the triangle is twice as long as the base. What are the base and the height of the triangle?
::三角形的区域是 24 英寸2 , 三角形的高度是基数的两倍。三角形的基数和高度是多少 ?

The length of a rectangle is 7 meters less than twice its width, and its area is $\begin{align*}660 \ m^2\end{align*}$ . What are the length and width of the rectangle?
::矩形的长度是宽度的两倍以下7米,其面积是660平方米。矩形的长度和宽度是多少?

The area of a circular disk is $\begin{align*}124 \ in^2\end{align*}$ . What is the circumference of the disk? $\begin{align*}(\text{Area} = \pi R^2, \text{Circumference} =2 \pi R)\end{align*}$ .
::圆盘区域为124英寸2,圆盘的环形范围是多少? (AreaR2,环形=2R)。

The volume of a cylinder is $\begin{align*}245 \ cm^3\end{align*}$ and the height of the cylinder is one third of the diameter of the base of the cylinder. The diameter of the cylinder is kept the same but the height of the cylinder is increased by 2 centimeters. What is the volume of the new cylinder? $\begin{align*}(\text{Volume} =\pi R^2 \cdot h)\end{align*}$
::气瓶的体积为245厘米3,气瓶的高度是气瓶底部直径的三分之一,气瓶的直径保持不变,但气瓶的高度增加2厘米。新气瓶的体积是多少? (VolumeR2h)

The height of a golf ball as it travels through the air is given by the equation $\begin{align*}h=-16t^2+256\end{align*}$ . Find the time when the ball is at a height of 120 feet.
::方程式 h16t2+256给出高尔夫球在空中飞行时的高度。找到球在120英尺高度的时间。

Review (Answers)
::回顾(答复)

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::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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