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شائع علم المثلثات >

tan(2sqrt(x)-3)=-1

الحلّ

tan (2 x - 3) = - 1

الحلّ

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

خطوات الحلّ

tan (2 x - 3) = - 1

tan (2 x - 3) = - 1 :

حلول عامّة لـ

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

2 x - 3 = \frac{3 π}{4} + πn

2 x - 3 = \frac{3 π}{4} + πn

2 x - 3 = \frac{3 π}{4} + πn

حلّ:

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} {n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}}

2 x - 3 = \frac{3 π}{4} + πn

4

اضرب الطرفين بـ

2 x \cdot 4 - 3 \cdot 4 = \frac{3 π}{4} \cdot 4 + πn \cdot 4

بسّط

8 x - 12 = 3 π + 4 πn

انقل

12

إلى الجانب الأيمن

8 x - 12 = 3 π + 4 πn

للطرفين

12

أضف

8 x - 12 + 12 = 3 π + 4 πn + 12

بسّط

8 x = 3 π + 4 πn + 12

8 x = 3 π + 4 πn + 12

8

اقسم الطرفين على

8 x = 3 π + 4 πn + 12

8

اقسم الطرفين على

\frac{8 x}{8} = \frac{3 π}{8} + \frac{4 πn}{8} + \frac{12}{8}

بسّط

\frac{8 x}{8} = \frac{3 π}{8} + \frac{4 πn}{8} + \frac{12}{8}

\frac{8 x}{8}

بسّط:

x

\frac{8 x}{8}

\frac{8}{8} = 1 :

اقسم الأعداد

= x

\frac{3 π}{8} + \frac{4 πn}{8} + \frac{12}{8}

بسّط:

\frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2}

\frac{3 π}{8} + \frac{4 πn}{8} + \frac{12}{8}

جمّع التعابير المتشابهة

= \frac{3 π}{8} + \frac{12}{8} + \frac{4 πn}{8}

\frac{12}{8}

اختزل:

\frac{3}{2}

\frac{12}{8}

4 :

إلغ العوامل المشتركة

= \frac{3}{2}

= \frac{3 π}{8} + \frac{3}{2} + \frac{4 πn}{8}

\frac{4 πn}{8}

اختزل:

\frac{πn}{2}

\frac{4 πn}{8}

4 :

إلغ العوامل المشتركة

= \frac{πn}{2}

= \frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2}

x = \frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2}

ربّع الطرفين:

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

x = \frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2}

(x)^{2} = (\frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2})^{2}

(x)^{2}

وسّع:

x

(x)^{2}

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= (x^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= x^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= x

(\frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2})^{2}

وسّع:

\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

(\frac{3 π}{8} + \frac{3}{2} + \frac{πn}{2})^{2}

\frac{3}{2} + \frac{πn}{2}

وحّد الكسور:

\frac{3 + πn}{2}

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

فعّل القانون

= \frac{3 + πn}{2}

= (\frac{3 π}{8} + \frac{πn + 3}{2})^{2}

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = \frac{3 π}{8}, b = \frac{3 + πn}{2}

= (\frac{3 π}{8})^{2} + 2 \cdot \frac{3 π}{8} \cdot \frac{3 + πn}{2} + (\frac{3 + πn}{2})^{2}

(\frac{3 π}{8})^{2} + 2 \cdot \frac{3 π}{8} \cdot \frac{3 + πn}{2} + (\frac{3 + πn}{2})^{2}

بسّط:

\frac{9 π ^{2}}{64} + \frac{9 π + 3 π ^{2} n}{8} + \frac{9 + 6 πn + π ^{2} n ^{2}}{4}

(\frac{3 π}{8})^{2} + 2 \cdot \frac{3 π}{8} \cdot \frac{3 + πn}{2} + (\frac{3 + πn}{2})^{2}

(\frac{3 π}{8})^{2} = \frac{9 π ^{2}}{64}

(\frac{3 π}{8})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{( 3 π ) ^{2}}{8 ^{2}}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

(3 π)^{2} = 3^{2} π^{2}

= \frac{3 ^{2} π ^{2}}{8 ^{2}}

بسّط

= \frac{9 π ^{2}}{64}

2 \cdot \frac{3 π}{8} \cdot \frac{3 + πn}{2} = \frac{9 π + 3 π ^{2} n}{8}

2 \cdot \frac{3 π}{8} \cdot \frac{3 + πn}{2}

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

:اضرب كسور

= \frac{3 π ( 3 + πn ) \cdot 2}{8 \cdot 2}

2 :

إلغ العوامل المشتركة

= \frac{3 π ( 3 + πn )}{8}

3 π (3 + πn)

وسٌع:

9 π + 3 π^{2} n

3 π (3 + πn)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = 3 π, b = 3, c = πn

= 3 π 3 + 3 ππn

= 3 \cdot 3 π + 3 ππn

3 \cdot 3 π + 3 ππn

بسّط:

9 π + 3 π^{2} n

3 \cdot 3 π + 3 ππn

3 \cdot 3 π = 9 π

3 \cdot 3 π

3 \cdot 3 = 9 :

اضرب الأعداد

= 9 π

3 ππn = 3 π^{2} n

3 ππn

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

ππ = π^{1 + 1}

= 3 π^{1 + 1} n

1 + 1 = 2 :

اجمع الأعداد

= 3 π^{2} n

= 9 π + 3 π^{2} n

= 9 π + 3 π^{2} n

= \frac{9 π + 3 π ^{2} n}{8}

(\frac{3 + πn}{2})^{2} = \frac{9 + 6 πn + π ^{2} n ^{2}}{4}

(\frac{3 + πn}{2})^{2}

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

:فعّل قانون القوى

= \frac{( 3 + πn ) ^{2}}{2 ^{2}}

(3 + πn)^{2} = 9 + 6 πn + π^{2} n^{2}

(3 + πn)^{2}

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = 3, b = πn

= 3^{2} + 2 \cdot 3 πn + (πn)^{2}

3^{2} + 2 \cdot 3 πn + (πn)^{2}

بسّط:

9 + 6 πn + π^{2} n^{2}

3^{2} + 2 \cdot 3 πn + (πn)^{2}

3^{2} = 9

= 9 + 2 \cdot 3 πn + (πn)^{2}

2 \cdot 3 = 6 :

اضرب الأعداد

= 9 + 6 πn + (πn)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 9 + 6 πn + π^{2} n^{2}

= 9 + 6 πn + π^{2} n^{2}

= \frac{9 + 6 πn + π ^{2} n ^{2}}{2 ^{2}}

2^{2} = 4

= \frac{9 + 6 πn + π ^{2} n ^{2}}{4}

= \frac{9 π ^{2}}{64} + \frac{3 π ^{2} n + 9 π}{8} + \frac{π ^{2} n ^{2} + 6 πn + 9}{4}

= \frac{9 π ^{2}}{64} + \frac{9 π + 3 π ^{2} n}{8} + \frac{9 + 6 πn + π ^{2} n ^{2}}{4}

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

: استخدم ميزات الكسور التالية

\frac{9 π + 3 π ^{2} n}{8} = \frac{9 π}{8} + \frac{3 π ^{2} n}{8}

= \frac{9 π ^{2}}{64} + \frac{9 π}{8} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2} + 6 πn + 9}{4}

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

: استخدم ميزات الكسور التالية

\frac{9 + 6 πn + π ^{2} n ^{2}}{4} = \frac{9}{4} + \frac{6 πn}{4} + \frac{π ^{2} n ^{2}}{4}

= \frac{9 π ^{2}}{64} + \frac{9 π}{8} + \frac{3 π ^{2} n}{8} + \frac{9}{4} + \frac{6 πn}{4} + \frac{π ^{2} n ^{2}}{4}

جمّع التعابير المتشابهة

= \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{6 πn}{4} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

\frac{6 πn}{4}

اختزل:

\frac{3 πn}{2}

\frac{6 πn}{4}

2 :

إلغ العوامل المشتركة

= \frac{3 πn}{2}

= \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

افحص الإجبات:

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} {n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}}

للتحقّق من دقّة الحلول

2 x - 3 = \frac{3 π}{4} + πn

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}} \Leftarrow 2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 3 = \frac{3 π}{4} + πn : x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

عوّض

2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 3 = \frac{3 π}{4} + πn

4

اضرب الطرفين بـ

2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} \cdot 4 - 3 \cdot 4 = \frac{3 π}{4} \cdot 4 + πn \cdot 4

بسّط

8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 12 = 3 π + 4 πn

Remove square roots

8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 12 = 3 π + 4 πn

للطرفين

12

أضف

8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 12 + 12 = 3 π + 4 πn + 12

بسّط

8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} = 3 π + 4 πn + 12

ربّع الطرفين:

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 12 = 3 π + 4 πn

(8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{2} = (3 π + 4 πn + 12)^{2}

(8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{2}

وسّع:

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

(8 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 8^{2} (\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{2}

(\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{2} : \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

a = a^{\frac{1}{2}}

:فعْل قانون الجذور

= ((\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{\frac{1}{2}})^{2}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= (\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})^{\frac{1}{2} \cdot 2}

\frac{1}{2} \cdot 2 = 1

\frac{1}{2} \cdot 2

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{1 \cdot 2}{2}

2 :

إلغ العوامل المشتركة

= 1

= \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

= 8^{2} (\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})

8^{2} = 64

= 64 (\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})

64 (\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})

وسّع:

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

64 (\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4})

\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

بسّط:

\frac{π ^{2} n ^{2} + 9}{4} + \frac{3 π ^{2} n + 9 π}{8} + \frac{3 πn}{2} + \frac{9 π ^{2}}{64}

\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

\frac{9}{4} + \frac{π ^{2} n ^{2}}{4}

وحّد الكسور:

\frac{9 + π ^{2} n ^{2}}{4}

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

فعّل القانون

= \frac{9 + π ^{2} n ^{2}}{4}

= \frac{π ^{2} n ^{2} + 9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8}

\frac{9 π}{8} + \frac{3 π ^{2} n}{8}

وحّد الكسور:

\frac{9 π + 3 π ^{2} n}{8}

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

فعّل القانون

= \frac{9 π + 3 π ^{2} n}{8}

= \frac{π ^{2} n ^{2} + 9}{4} + \frac{3 π ^{2} n + 9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2}

= 64 (\frac{π ^{2} n ^{2} + 9}{4} + \frac{3 π ^{2} n + 9 π}{8} + \frac{3 πn}{2} + \frac{9 π ^{2}}{64})

فعّل قانون ضرب الأقواس

= 64 \cdot \frac{9 + π ^{2} n ^{2}}{4} + 64 \cdot \frac{9 π + 3 π ^{2} n}{8} + 64 \cdot \frac{9 π ^{2}}{64} + 64 \cdot \frac{3 πn}{2}

64 \cdot \frac{9 + π ^{2} n ^{2}}{4} + 64 \cdot \frac{9 π + 3 π ^{2} n}{8} + 64 \cdot \frac{9 π ^{2}}{64} + 64 \cdot \frac{3 πn}{2}

بسّط:

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

64 \cdot \frac{9 + π ^{2} n ^{2}}{4} + 64 \cdot \frac{9 π + 3 π ^{2} n}{8} + 64 \cdot \frac{9 π ^{2}}{64} + 64 \cdot \frac{3 πn}{2}

64 \cdot \frac{9 + π ^{2} n ^{2}}{4} = 16 π^{2} n^{2} + 144

64 \cdot \frac{9 + π ^{2} n ^{2}}{4}

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{( 9 + π ^{2} n ^{2} ) \cdot 64}{4}

\frac{64}{4} = 16 :

اقسم الأعداد

= 16 (π^{2} n^{2} + 9)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = 16, b = π^{2} n^{2}, c = 9

= 16 π^{2} n^{2} + 16 \cdot 9

16 \cdot 9 = 144 :

اضرب الأعداد

= 16 π^{2} n^{2} + 144

64 \cdot \frac{9 π + 3 π ^{2} n}{8} = 24 π^{2} n + 72 π

64 \cdot \frac{9 π + 3 π ^{2} n}{8}

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{( 9 π + 3 π ^{2} n ) \cdot 64}{8}

\frac{64}{8} = 8 :

اقسم الأعداد

= 8 (3 π^{2} n + 9 π)

a (b + c) = ab + a c

: افتح أقواس بالاستعانة بـ

a = 8, b = 3 π^{2} n, c = 9 π

= 8 \cdot 3 π^{2} n + 8 \cdot 9 π

8 \cdot 3 π^{2} n + 8 \cdot 9 π

بسّط:

24 π^{2} n + 72 π

8 \cdot 3 π^{2} n + 8 \cdot 9 π

8 \cdot 3 = 24 :

اضرب الأعداد

= 24 π^{2} n + 8 \cdot 9 π

8 \cdot 9 = 72 :

اضرب الأعداد

= 24 π^{2} n + 72 π

= 24 π^{2} n + 72 π

64 \cdot \frac{9 π ^{2}}{64} = 9 π^{2}

64 \cdot \frac{9 π ^{2}}{64}

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{9 π ^{2} \cdot 64}{64}

64 :

إلغ العوامل المشتركة

= 9 π^{2}

64 \cdot \frac{3 πn}{2} = 96 πn

64 \cdot \frac{3 πn}{2}

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

:اضرب كسور

= \frac{3 πn \cdot 64}{2}

3 \cdot 64 = 192 :

اضرب الأعداد

= \frac{192 πn}{2}

\frac{192}{2} = 96 :

اقسم الأعداد

= 96 πn

= 16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

= 16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

= 16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

(3 π + 4 πn + 12)^{2}

وسّع:

9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

(3 π + 4 πn + 12)^{2}

(3 π + 4 πn + 12)^{2} = (3 π + 4 πn + 12) (3 π + 4 πn + 12)

= (3 π + 4 πn + 12) (3 π + 4 πn + 12)

(3 π + 4 πn + 12) (3 π + 4 πn + 12)

وسٌع:

9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

(3 π + 4 πn + 12) (3 π + 4 πn + 12)

فعّل قانون ضرب الأقواس

= 3 π 3 π + 3 π 4 πn + 3 π 12 + 4 πn \cdot 3 π + 4 πn \cdot 4 πn + 4 πn \cdot 12 + 12 \cdot 3 π + 12 \cdot 4 πn + 12 \cdot 12

= 3 \cdot 3 ππ + 3 \cdot 4 ππn + 3 \cdot 12 π + 4 \cdot 3 ππn + 4 \cdot 4 ππnn + 4 \cdot 12 πn + 12 \cdot 3 π + 12 \cdot 4 πn + 12 \cdot 12

3 \cdot 3 ππ + 3 \cdot 4 ππn + 3 \cdot 12 π + 4 \cdot 3 ππn + 4 \cdot 4 ππnn + 4 \cdot 12 πn + 12 \cdot 3 π + 12 \cdot 4 πn + 12 \cdot 12

بسّط:

9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

3 \cdot 3 ππ + 3 \cdot 4 ππn + 3 \cdot 12 π + 4 \cdot 3 ππn + 4 \cdot 4 ππnn + 4 \cdot 12 πn + 12 \cdot 3 π + 12 \cdot 4 πn + 12 \cdot 12

3 \cdot 12 π + 12 \cdot 3 π = 2 \cdot 12 \cdot 3 π :

اجمع العناصر المتشابهة

= 3 \cdot 3 ππ + 3 \cdot 4 ππn + 2 \cdot 12 \cdot 3 π + 4 \cdot 3 ππn + 4 \cdot 4 ππnn + 4 \cdot 12 πn + 12 \cdot 4 πn + 12 \cdot 12

4 \cdot 12 πn + 12 \cdot 4 πn = 2 \cdot 12 \cdot 4 πn :

اجمع العناصر المتشابهة

= 3 \cdot 3 ππ + 3 \cdot 4 ππn + 2 \cdot 12 \cdot 3 π + 4 \cdot 3 ππn + 4 \cdot 4 ππnn + 2 \cdot 12 \cdot 4 πn + 12 \cdot 12

3 \cdot 4 ππn + 4 \cdot 3 ππn = 2 \cdot 4 \cdot 3 ππn :

اجمع العناصر المتشابهة

= 3 \cdot 3 ππ + 2 \cdot 4 \cdot 3 ππn + 2 \cdot 12 \cdot 3 π + 4 \cdot 4 ππnn + 2 \cdot 12 \cdot 4 πn + 12 \cdot 12

3 \cdot 3 ππ = 9 π^{2}

3 \cdot 3 ππ

3 \cdot 3 = 9 :

اضرب الأعداد

= 9 ππ

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

ππ = π^{1 + 1}

= 9 π^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= 9 π^{2}

2 \cdot 4 \cdot 3 ππn = 24 π^{2} n

2 \cdot 4 \cdot 3 ππn

2 \cdot 4 \cdot 3 = 24 :

اضرب الأعداد

= 24 ππn

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

ππ = π^{1 + 1}

= 24 π^{1 + 1} n

1 + 1 = 2 :

اجمع الأعداد

= 24 π^{2} n

2 \cdot 12 \cdot 3 π = 72 π

2 \cdot 12 \cdot 3 π

2 \cdot 12 \cdot 3 = 72 :

اضرب الأعداد

= 72 π

4 \cdot 4 ππnn = 16 π^{2} n^{2}

4 \cdot 4 ππnn

4 \cdot 4 = 16 :

اضرب الأعداد

= 16 ππnn

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

nn = n^{1 + 1}

= 16 ππ n^{1 + 1}

1 + 1 = 2 :

اجمع الأعداد

= 16 ππ n^{2}

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

ππ = π^{1 + 1}

= 16 π^{1 + 1} n^{2}

1 + 1 = 2 :

اجمع الأعداد

= 16 π^{2} n^{2}

2 \cdot 12 \cdot 4 πn = 96 πn

2 \cdot 12 \cdot 4 πn

2 \cdot 12 \cdot 4 = 96 :

اضرب الأعداد

= 96 πn

12 \cdot 12 = 144

12 \cdot 12

12 \cdot 12 = 144 :

اضرب الأعداد

= 144

= 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

= 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

= 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

حلّ:

يتحقّق لكلّ

n

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144

من الطرفين

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn

اطرح

16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn - (16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn) = 9 π^{2} + 24 π^{2} n + 72 π + 16 π^{2} n^{2} + 96 πn + 144 - (16 π^{2} n^{2} + 144 + 24 π^{2} n + 72 π + 9 π^{2} + 96 πn)

بسّط

0 = 0

يتساوى الطرفان

يتحقّق لكلّ n

يتحقّق لكلّ n

افحص الإجبات:

n < - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

خطأ

, n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

خطأ

, n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

صحيح

2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 3 = \frac{3 π}{4} + πn

ادمج المجالات مع الحلول

يتحقّق لكلّ n

جد المقاطع للدالّة:

n < - \frac{96 π + 24 π ^{2}}{32 π ^{2}}, n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}, n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 3 = \frac{3 π}{4} + πn

جدّ الحلول للجذور الزوجيّة

\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} = 0

حلّ:

n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

\frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} = 0

Find Least Common Multiplier of

4, 8, 64, 2 : 64

4, 8, 64, 2

المضاعف المشترك الأصغر

4

تحليل لعوامل أوّليّة لـ:

2 \cdot 2

4

4 = 2 \cdot 2, 2

ينقسم على

4

= 2 \cdot 2

8

تحليل لعوامل أوّليّة لـ:

2 \cdot 2 \cdot 2

8

8 = 4 \cdot 2, 2

ينقسم على

8

= 2 \cdot 4

4 = 2 \cdot 2, 2

ينقسم على

4

= 2 \cdot 2 \cdot 2

64

تحليل لعوامل أوّليّة لـ:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

64

64 = 32 \cdot 2, 2

ينقسم على

64

= 2 \cdot 32

32 = 16 \cdot 2, 2

ينقسم على

32

= 2 \cdot 2 \cdot 16

16 = 8 \cdot 2, 2

ينقسم على

16

= 2 \cdot 2 \cdot 2 \cdot 8

8 = 4 \cdot 2, 2

ينقسم على

8

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4

4 = 2 \cdot 2, 2

ينقسم على

4

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2

تحليل لعوامل أوّليّة لـ:

2

2

هو عدد أوّليّ لذلك لا يمكن تحليله لعوامل أوّليّة

2

= 2

احسب عدد مركّب من عوامل أوّليّة تظهر بواحد من التعابير التالية على الأقلّ

4, 8, 64, 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 64 :

اضرب الأعداد

= 64

64 =

اضرب بالمضاعف المشترك الأصغر

\frac{9}{4} \cdot 64 + \frac{9 π}{8} \cdot 64 + \frac{9 π ^{2}}{64} \cdot 64 + \frac{3 πn}{2} \cdot 64 + \frac{3 π ^{2} n}{8} \cdot 64 + \frac{π ^{2} n ^{2}}{4} \cdot 64 = 0 \cdot 64

بسّط

16 π^{2} n^{2} + (96 π + 24 π^{2}) n + 144 + 72 π + 9 π^{2} = 0

حلّ بالاستعانة بالصيغة التربيعيّة

16 π^{2} n^{2} + (96 π + 24 π^{2}) n + 144 + 72 π + 9 π^{2} = 0

الصيغة لحلّ المعادلة التربيعيّة

a = 16 π^{2}, b = 96 π + 24 π^{2}, c = 144 + 72 π + 9 π^{2}

لـ

n_{1, 2} = \frac{- ( 96 π + 24 π ^{2} ) \pm ( 96 π + 24 π ^{2} ) ^{2} - 4 \cdot 16 π ^{2} ( 144 + 72 π + 9 π ^{2} )}{2 \cdot 16 π ^{2}}

n_{1, 2} = \frac{- ( 96 π + 24 π ^{2} ) \pm ( 96 π + 24 π ^{2} ) ^{2} - 4 \cdot 16 π ^{2} ( 144 + 72 π + 9 π ^{2} )}{2 \cdot 16 π ^{2}}

(96 π + 24 π^{2})^{2} - 4 \cdot 16 π^{2} (144 + 72 π + 9 π^{2}) = 0

(96 π + 24 π^{2})^{2} - 4 \cdot 16 π^{2} (144 + 72 π + 9 π^{2})

4 \cdot 16 = 64 :

اضرب الأعداد

= (24 π^{2} + 96 π)^{2} - 64 π^{2} (9 π^{2} + 72 π + 144)

(96 π + 24 π^{2})^{2} : 9216 π^{2} + 4608 π^{3} + 576 π^{4}

(a + b)^{2} = a^{2} + 2 ab + b^{2}

:فعّل صيغة الضرب المختصر

a = 96 π, b = 24 π^{2}

= (96 π)^{2} + 2 \cdot 96 π 24 π^{2} + (24 π^{2})^{2}

(96 π)^{2} + 2 \cdot 96 π 24 π^{2} + (24 π^{2})^{2}

بسّط:

9216 π^{2} + 4608 π^{3} + 576 π^{4}

(96 π)^{2} + 2 \cdot 96 π 24 π^{2} + (24 π^{2})^{2}

(96 π)^{2} = 9216 π^{2}

(96 π)^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 9 6^{2} π^{2}

9 6^{2} = 9216

= 9216 π^{2}

2 \cdot 96 π 24 π^{2} = 4608 π^{3}

2 \cdot 96 π 24 π^{2}

2 \cdot 96 \cdot 24 = 4608 :

اضرب الأعداد

= 4608 π^{2} π

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

π π^{2} = π^{1 + 2}

= 4608 π^{1 + 2}

1 + 2 = 3 :

اجمع الأعداد

= 4608 π^{3}

(24 π^{2})^{2} = 576 π^{4}

(24 π^{2})^{2}

(a \cdot b)^{n} = a^{n} b^{n}

:فعّل قانون القوى

= 2 4^{2} (π^{2})^{2}

(π^{2})^{2} : π^{4}

(a^{b})^{c} = a^{b c}

:فعّل قانون القوى

= π^{2 \cdot 2}

2 \cdot 2 = 4 :

اضرب الأعداد

= π^{4}

= 2 4^{2} π^{4}

2 4^{2} = 576

= 576 π^{4}

= 9216 π^{2} + 4608 π^{3} + 576 π^{4}

= 9216 π^{2} + 4608 π^{3} + 576 π^{4}

= 9216 π^{2} + 4608 π^{3} + 576 π^{4} - 64 π^{2} (144 + 72 π + 9 π^{2})

- 64 π^{2} (144 + 72 π + 9 π^{2})

وسٌع:

- 9216 π^{2} - 4608 π^{3} - 576 π^{4}

- 64 π^{2} (144 + 72 π + 9 π^{2})

فعّل قانون ضرب الأقواس

= (- 64 π^{2}) \cdot 144 + (- 64 π^{2}) \cdot 72 π + (- 64 π^{2}) \cdot 9 π^{2}

فعّل قوانين سالب-موجب

+ (- a) = - a

= - 64 \cdot 144 π^{2} - 64 \cdot 72 π^{2} π - 64 \cdot 9 π^{2} π^{2}

- 64 \cdot 144 π^{2} - 64 \cdot 72 π^{2} π - 64 \cdot 9 π^{2} π^{2}

بسّط:

- 9216 π^{2} - 4608 π^{3} - 576 π^{4}

- 64 \cdot 144 π^{2} - 64 \cdot 72 π^{2} π - 64 \cdot 9 π^{2} π^{2}

64 \cdot 144 π^{2} = 9216 π^{2}

64 \cdot 144 π^{2}

64 \cdot 144 = 9216 :

اضرب الأعداد

= 9216 π^{2}

64 \cdot 72 π^{2} π = 4608 π^{3}

64 \cdot 72 π^{2} π

64 \cdot 72 = 4608 :

اضرب الأعداد

= 4608 π^{2} π

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

π^{2} π = π^{2 + 1}

= 4608 π^{2 + 1}

2 + 1 = 3 :

اجمع الأعداد

= 4608 π^{3}

64 \cdot 9 π^{2} π^{2} = 576 π^{4}

64 \cdot 9 π^{2} π^{2}

64 \cdot 9 = 576 :

اضرب الأعداد

= 576 π^{2} π^{2}

a^{b} \cdot a^{c} = a^{b + c}

:فعّل قانون القوى

π^{2} π^{2} = π^{2 + 2}

= 576 π^{2 + 2}

2 + 2 = 4 :

اجمع الأعداد

= 576 π^{4}

= - 9216 π^{2} - 4608 π^{3} - 576 π^{4}

= - 9216 π^{2} - 4608 π^{3} - 576 π^{4}

= 9216 π^{2} + 4608 π^{3} + 576 π^{4} - 9216 π^{2} - 4608 π^{3} - 576 π^{4}

9216 π^{2} + 4608 π^{3} + 576 π^{4} - 9216 π^{2} - 4608 π^{3} - 576 π^{4}

بسّط:

0

9216 π^{2} + 4608 π^{3} + 576 π^{4} - 9216 π^{2} - 4608 π^{3} - 576 π^{4}

جمّع التعابير المتشابهة

= 576 π^{4} - 576 π^{4} + 4608 π^{3} - 4608 π^{3} + 9216 π^{2} - 9216 π^{2}

9216 π^{2} - 9216 π^{2} = 0 :

اجمع العناصر المتشابهة

= 576 π^{4} - 576 π^{4} + 4608 π^{3} - 4608 π^{3}

4608 π^{3} - 4608 π^{3} = 0 :

اجمع العناصر المتشابهة

= 576 π^{4} - 576 π^{4}

576 π^{4} - 576 π^{4} = 0 :

اجمع العناصر المتشابهة

= 0

= 0

n_{1, 2} = \frac{- ( 96 π + 24 π ^{2} ) \pm 0}{2 \cdot 16 π ^{2}}

n = \frac{- ( 96 π + 24 π ^{2} )}{2 \cdot 16 π ^{2}}

\frac{- ( 96 π + 24 π ^{2} )}{2 \cdot 16 π ^{2}} = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

\frac{- ( 96 π + 24 π ^{2} )}{2 \cdot 16 π ^{2}}

2 \cdot 16 = 32 :

اضرب الأعداد

= \frac{- ( 24 π ^{2} + 96 π )}{32 π ^{2}}

\frac{- a}{b} = - \frac{a}{b}

: استخدم ميزات الكسور التالية

= - \frac{( 96 π + 24 π ^{2} )}{32 π ^{2}}

(a) = a

:احذف الأقواس

= - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

حلّ المعادلة التربيعيّة هو

n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

المقاطع معرّفة حول الصفر

n < - \frac{96 π + 24 π ^{2}}{32 π ^{2}}, n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}, n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

وحّد القطع مع مجال تعريف الدالّة

n < - \frac{96 π + 24 π ^{2}}{32 π ^{2}}, n = - \frac{96 π + 24 π ^{2}}{32 π ^{2}}, n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

للتحقّق من دقّة الحلول

2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 3 = \frac{3 π}{4} + πn

عوّض الحلول في

إلغي الحلول التي تعطي قضيّة كذب

قضيّة كذب

\Leftarrow 2 \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} - 3 \neq = \frac{3 π}{4} + πn

n < - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

عوّض

الحل للمعادلة هو

n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}

الحل للمعادلة هو

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4} {n > - \frac{96 π + 24 π ^{2}}{32 π ^{2}}}

x = \frac{9}{4} + \frac{9 π}{8} + \frac{9 π ^{2}}{64} + \frac{3 πn}{2} + \frac{3 π ^{2} n}{8} + \frac{π ^{2} n ^{2}}{4}

رسم

أمثلة شائعة

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