Course: 10.4 使用平根来解析赤道等量

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使用平方根来解析二次方等量
Use Square Roots to Solve Quadratic Equations
::使用平方根来解析二次方等量

So far you know how to by factoring. However, this method works only if a quadratic polynomial can be factored. In the real world, most quadratics can’t be factored, so now we’ll start to learn other methods we can use to solve them. In this Concept, we’ll examine equations in which we can take the square root of both sides of the equation in order to arrive at the result.
::到目前为止,你知道如何通过计数法。然而,这种方法只有在四边多面体因素因素因素的情况下才能奏效。在现实世界中,大多数四方体因素无法计算,因此我们现在将开始学习其他方法,我们可以用这些方法来解决这些问题。在这个概念中,我们将研究方程式的方程式,我们可以从方程式的两侧的平方根中找到正方程式,以便得出结果。

Solve Quadratic Equations Involving Perfect Squares
::涉及完美广场的溶解二次赤道等同

Let’s first examine quadratic equations of the type
::让我们首先研究这种类型的二次方程

$\begin{aligned} x^{2} - c = 0 \end{aligned}$

::x2 - c=0

We can solve this equation by isolating the $\begin{aligned} x^{2} \end{aligned}$ term : $\begin{aligned} x^{2} = c \end{aligned}$
::我们可以通过分离 x2 术语: x2= c 来解析此方程式

Once the $\begin{aligned} x^{2} \end{aligned}$ term is isolated we can take the square root of both sides of the equation. Remember that when we take the square root we get two answers: the positive square root and the negative square root:
::当 x2 术语被孤立时, 我们可以选择方程式两侧的平方根。记住当我们选择平方根时, 我们得到两个答案: 正方根和负方根 :

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

::x=candxc

Often this is written as $\begin{aligned} x = \pm \sqrt{c} \end{aligned}$ .
::通常将此写成 xc。

Solving for Unknown Values
::解决未知值

1. Solve the following quadratic equations:
::1. 解决下列二次方程:

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

Isolate the $\begin{aligned} x^{2} \end{aligned}$ : $\begin{aligned} x^{2} = 4 \end{aligned}$
::孤立 x2 : x2=4

Take the square root of both sides: $\begin{aligned} x = \sqrt{4} \end{aligned}$ and $\begin{aligned} x = - \sqrt{4} \end{aligned}$
::以两侧的平方根为平方根: x=4 和 x4

The solutions are $\begin{aligned} x = 2 \end{aligned}$ and $\begin{aligned} x = - 2 \end{aligned}$ .
::解决方案为 x=2andx @% 2 。

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

Isolate the $\begin{aligned} x^{2} \end{aligned}$ : $\begin{aligned} x^{2} = 25 \end{aligned}$
::隔离 x2: x2=25

Take the square root of both sides: $\begin{aligned} x = \sqrt{25} \end{aligned}$ and $\begin{aligned} x = - \sqrt{25} \end{aligned}$
::以两侧的平方根为平方根: x=25 和 x25

The solutions are $\begin{aligned} x = 5 \end{aligned}$ and $\begin{aligned} x = - 5 \end{aligned}$ .
::解决方案为 x=5 和 x=5 。

We can also find the solution using the square root when the $\begin{aligned} x^{2} \end{aligned}$ term is multiplied by a constant—in other words, when the equation takes the form
::当 x2 术语乘以一个恒定值时,也就是当方程式以表单形式出现时,我们也可以找到使用平方根的解决方案。

$\begin{aligned} a x^{2} - c = 0 \end{aligned}$

::ax2 - c=0

We just have to isolate the $\begin{aligned} x^{2} \end{aligned}$ :
::我们必须分离x2:

$\begin{aligned} a x^{2} & = b \\ x^{2} & = \frac{b}{a} \end{aligned}$

::ax2=bx2=ba

Then we can take the square root of both sides of the equation:
::这样我们就能从两边的平方根中找出两边的平方根:

$\begin{aligned} x = \sqrt{\frac{b}{a}} and x = - \sqrt{\frac{b}{a}} \end{aligned}$

::x=baandxba

Often this is written as: $\begin{aligned} x = \pm \sqrt{\frac{b}{a}} \end{aligned}$ .
::通常写为:xba。

2. Solve the following quadratic equations.
::2. 解决以下四方方程式。

a) $\begin{aligned} 9 x^{2} - 16 = 0 \end{aligned}$
:a) 9x2-16=0

Isolate the $\begin{aligned} x^{2} \end{aligned}$ :
::隔离 x2 :

$\begin{aligned} 9 x^{2} & = 16 \\ x^{2} & = \frac{16}{9} \end{aligned}$

::9x2=16x2=169

Take the square root of both sides: $\begin{aligned} x = \sqrt{\frac{16}{9}} \end{aligned}$ and $\begin{aligned} x = - \sqrt{\frac{16}{9}} \end{aligned}$
::以两侧的平方根: x=169 和 x169

Answer: $\begin{aligned} x = \frac{4}{3} \end{aligned}$ and $\begin{aligned} x = - \frac{4}{3} \end{aligned}$
::答复:x=43和x__________________________________________________________________________________________________________________________________________________________

b) $\begin{aligned} 81 x^{2} - 1 = 0 \end{aligned}$
:b) 81x2-1=0

Isolate the $\begin{aligned} x^{2} \end{aligned}$ :
::隔离 x2 :

$\begin{aligned} 81 x^{2} & = 1 \\ x^{2} & = \frac{1}{81} \end{aligned}$

::81x2=1x2=181

Take the square root of both sides: $\begin{aligned} x = \sqrt{\frac{1}{81}} \end{aligned}$ and $\begin{aligned} x = - \sqrt{\frac{1}{81}} \end{aligned}$
::以两边的平方根为平方根: x=181 和 x181

Answer: $\begin{aligned} x = \frac{1}{9} \end{aligned}$ and $\begin{aligned} x = - \frac{1}{9} \end{aligned}$
::答复:x=19和x19

As you’ve seen previously, some quadratic equations have no real solutions.
::有些二次方程式没有真正的解决方案。

3. Solve the following quadratic equations.
::3. 解决以下四方方程式。

a) $\begin{aligned} x^{2} + 1 = 0 \end{aligned}$
::a) x2+1=0

Isolate the $\begin{aligned} x^{2} \end{aligned}$ : $\begin{aligned} x^{2} = - 1 \end{aligned}$
::孤立 x2 : x2 @% 1

Take the square root of both sides: $\begin{aligned} x = \sqrt{- 1} \end{aligned}$ and $\begin{aligned} x = - \sqrt{- 1} \end{aligned}$
::以两侧的平方根为平方根: x1 和 x1

Square roots of negative numbers do not give real number results, so there are no real solutions to this equation.
::负数的平方根并不产生实际数字结果,因此这一方程式没有真正的解决办法。

b) $\begin{aligned} 4 x^{2} + 9 = 0 \end{aligned}$
::b) 4x2+9=0

Isolate the $\begin{aligned} x^{2} \end{aligned}$ :
::隔离 x2 :

$\begin{aligned} 4 x^{2} & = - 9 \\ x^{2} & = - \frac{9}{4} \end{aligned}$

::4x2%9x294

Take the square root of both sides: $\begin{aligned} x = \sqrt{- \frac{9}{4}} \end{aligned}$ and $\begin{aligned} x = - \sqrt{- \frac{9}{4}} \end{aligned}$
::以两侧的平方根为平方根: x94 和 x94

There are no real solutions.
::没有真正的解决办法。

We can also use the square root function in some quadratic equations where both sides of an equation are perfect squares. This is true if an equation is of this form:
::在某方程的两面都是完美的方形的二次方程中,我们也可以使用平方根函数。

$\begin{aligned} (x - 2)^{2} = 9 \end{aligned}$

:x-2)2=9

Both sides of the equation are perfect squares. We take the square root of both sides and end up with two equations: $\begin{aligned} x - 2 = 3 \end{aligned}$ and $\begin{aligned} x - 2 = - 3 \end{aligned}$ .
::等式的两边都是完美的正方形。我们选择两边的平方根, 最后两个方程式: x-2=3 和 x- 2\\\ 3 。

Solving both equations gives us $\begin{aligned} x = 5 \end{aligned}$ and $\begin{aligned} x = - 1 \end{aligned}$ .
::解析两个方程式时, x=5 和 x1 给我们带来 x=5 和 x1 。

4. Solve the following quadratic equations.
::4. 解决以下四方方程式。

a) $\begin{aligned} (x - 1)^{2} = 4 \end{aligned}$
::a) (x-1)2=4

$\begin{aligned} Take the square root of both sides : & x - 1 & = 2 and x - 1 = - 2 \\ Solve each equation : & x & = 3 and x = - 1 \end{aligned}$

::取取两边的平方根 : x- 1=2 和 x- 1\\\\\\\\\\ 2Solve 每一个方程式: x=3 和 x\\\ 1

Answer: $\begin{aligned} x = 3 \end{aligned}$ and $\begin{aligned} x = - 1 \end{aligned}$
::答复:x=3和x1

b) $\begin{aligned} (x + 3)^{2} = 1 \end{aligned}$
::b) (x+3)2=1

$\begin{aligned} Take the square root of both sides : & x + 3 & = 1 and x + 3 = - 1 \\ Solve each equation : & x & = - 2 and x = - 4 \end{aligned}$

::取取两边的平方根 : x+3=1 和 x+3\\\\\\\1Solve 每一个方程式: x%2 和 x4

Answer: $\begin{aligned} x = - 2 \end{aligned}$ and $\begin{aligned} x = - 4 \end{aligned}$
::答复:x%2和x%4

It might be necessary to factor the right-hand side of the equation as a perfect square before applying the method outlined above.
::在采用上述方法之前,也许有必要将方程式的右侧作为完美的平方因素加以考虑。

Examples
::实例

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

Example 1
::例1

$\begin{aligned} x^{2} + 8 x + 16 = 25 \end{aligned}$
::x2+8x+16=25

$\begin{aligned} Factor the right-hand-side : & x^{2} + 8 x + 16 = (x + 4)^{2} so (x + 4)^{2} = 25 \\ Take the square root of both sides : & x + 4 = 5 and x + 4 = - 5 \\ Solve each equation : & x = 1 and x = - 9 \end{aligned}$

::右侧: x2+8x+16=( x+4)2so( x+4)2=25 乘以两侧的平方根: x+4=5 和 x+4=5 5Solve 每一个方程式: x=1 和 x9

Answer: $\begin{aligned} x = 1 \end{aligned}$ and $\begin{aligned} x = - 9 \end{aligned}$
::答复:x=1和x9

Example 2
::例2

$\begin{aligned} 4 x^{2} - 20 x + 25 = 9 \end{aligned}$
::4x2-20x+25=9

$\begin{aligned} Factor the right-hand-side : & 4 x^{2} - 20 x + 25 = (2 x - 5)^{2} so (2 x - 5)^{2} = 9 \\ Take the square root of both sides : & 2 x - 5 = 3 and 2 x - 5 = - 3 \\ Solve each equation : & 2 x = 8 and 2 x = 2 \end{aligned}$

::右侧: 4x2 - 20x+25=( 2x - 5) 2so( 2x - 5) 2= 9 乘以两侧的平方根: 2x - 5=3 和 2x - 5=3 3Solve 每一个方程式: 2x=8 和 2x=2

Answer: $\begin{aligned} x = 4 \end{aligned}$ and $\begin{aligned} x = 1 \end{aligned}$
::答复:x=4和x=1

Review
::回顾

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

$\begin{aligned} x^{2} - 1 = 0 \end{aligned}$
::x2 - 1=0

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

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

$\begin{aligned} 9 x^{2} - 1 = 0 \end{aligned}$
::9x2 - 1=0

$\begin{aligned} 4 x^{2} - 49 = 0 \end{aligned}$
::4x2-49=0

$\begin{aligned} 64 x^{2} - 9 = 0 \end{aligned}$
::64x2- 9=0

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

$\begin{aligned} 25 x^{2} - 36 = 0 \end{aligned}$
::25x2-36=0

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

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

$\begin{aligned} x^{2} - 36 = 0 \end{aligned}$
::x2-36=0

$\begin{aligned} 16 x^{2} - 49 = 0 \end{aligned}$
::16x2-49=0

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

$\begin{aligned} (x + 5)^{2} = 16 \end{aligned}$
:x+5)2=16

$\begin{aligned} (2 x - 1)^{2} - 4 = 0 \end{aligned}$
:2x-1)2-4=0

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

$\begin{aligned} (x - 3)^{2} + 25 = 0 \end{aligned}$
:x-3)2+25=0

$\begin{aligned} x^{2} - 10 x + 25 = 9 \end{aligned}$
::x2 - 10x+25=9

$\begin{aligned} x^{2} + 18 x + 81 = 1 \end{aligned}$
::x2+18x+81=1

$\begin{aligned} 4 x^{2} - 12 x + 9 = 16 \end{aligned}$
::4x2 - 12x+9=16

$\begin{aligned} 2 (x + 3)^{2} = 8 \end{aligned}$
::2(x+3)2=8

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

使用平方根来解析二次方等量

Use Square Roots to Solve Quadratic Equations ::使用平方根来解析二次方等量

Solve Quadratic Equations Involving Perfect Squares ::涉及完美广场的溶解二次赤道等同

Solving for Unknown Values ::解决未知值

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Review ::回顾

Review (Answers) ::回顾(答复)

Use Square Roots to Solve Quadratic Equations
::使用平方根来解析二次方等量

Solve Quadratic Equations Involving Perfect Squares
::涉及完美广场的溶解二次赤道等同

Solving for Unknown Values
::解决未知值

Examples
::实例

Example 1
::例1

Example 2
::例2

Review
::回顾

Review (Answers)
::回顾(答复)