Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 3 Herrega kutaa 9: Qixxattoota Hiikuu

Wanti fuula kana irra jiru hundi boqonnaa 3 kitaaba Herrega kutaa 9 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

5
Kaardii: 10
Gaaffii shaakalaa boqonnaa 3: 87–129
Fuula kitaabaa boqonnaa 3

Boqonnaan 3 maal haguuga

1.Revision of Linear Equation in One Variable
2.Systems of Linear Equations in Two Variables
3.Solving Non-linear Equations
4.Some Applications of Solving Equations

Yaadannoo gabaabaa boqonnaa 3

Kaardiiwwan fuula Herrega kutaa 9 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Seera Hima Walqixxaa Gatii QulqulluuSeera

Yoom akka itti fayyadamtu

Hima walqixxaa mallattoo gatii qulqulluu of keessaa qabu furuuf itti fayyadamaa.

Of eeggadhu

Gatiin qulqulluu adda bahe lakkoofsa negaatiiviitti qixxaata yoo ta'e, himichi yeruma sana furmaata hin qabu.

Grade 9 Mathematics, fuula 87-129 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Diskiriiminaantii fi Amala HundeeffamaaGabatee wabiiD = b^2 - 4ac > 0 · D = b^2 - 4ac = 0 · D = b^2 - 4ac < 0

Gatii Diskiriiminaantii	Amala Hundeeffamaa	Baay'ina Hundee Qabatamaa
$D = b^2 - 4ac > 0$	Dhugaa fi adda addaa	$2$
$D = b^2 - 4ac = 0$	Dhugaa fi walqixa	$1$
$D = b^2 - 4ac < 0$	Hidhii dhugaa hin qabu	$0$

Yoom akka itti fayyadamtu

Hima walqixxaa kuwaadiraatikii utuu hin furin baay'ina fi gosa hundeeffama isaa beekuuf itti fayyadamaa.

Of eeggadhu

Diskiriiminaantiin zeeroo gadi yoo ta'e hundee qabatamaa hin qabu. Lakkoofsota qabatamaa keessatti hundee lakkoofsa negaatiivii hin fudhatinaa.

Grade 9 Mathematics, fuula 87-129 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Hundee Kijibaa Hima Walqixxaa RaadikaaliiDogoggora baratame

Yoom akka itti fayyadamtu

Hima walqixxaa jijjiiramaa hundee keessatti qabu yookiin ekspoonantii kFraction qabu yeroo furtan itti fayyadamaa.

Of eeggadhu

Garaa lamaan iskuweerii gochuun hundee sobaa fiduu danda'a. Hundee argame hunda hima walqixxaa jalqabaa keessatti mirkaneessuu qabdu.

Grade 9 Mathematics, fuula 87-129 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Foormulikii KuwaadiraatikiiFoormulaa

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

$x$: hundeeffama kuwaadiraatikii $none$
$a$: koo-efisiyantii duraa (a zeeroo kan hin taane) $none$
$b$: koo-efisiyantii gidduu $none$
$c$: lakkoofsa dhaabbataa $none$

Yoom akka itti fayyadamtu

Hundeeffama hima walqixxaa kuwaadiraatikii bifa dhaabbataa ax^2 + bx + c = 0 irratti argachuuf itti fayyadamaa.

Of eeggadhu

Hirmiinsi 2a lakkoofsa gubbaa guutuu -b dabalatee hira malee hundee qofa miti.

Grade 9 Mathematics, fuula 87-129 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Furmaata Sirna Hima Walqixxaa SaayiziiGabatee wabiia_1/a_2 b_1/b_2 wajjin wal qixa miti · a_1/a_2 = b_1/b_2 = c_1/c_2 · a_1/a_2 = b_1/b_2 c_1/c_2 wajjin wal qixa miti

Haala Reeshoo	Ramaddii Sirnaa	Baay'ina Furmaataa	Sararoota Joomitirii
$\frac{a_1}{a_2}$ $\frac{b_1}{b_2}$ wajjin wal qixa miti	Independent	Furmaata addaa tokko	Walqaxxaamuraa (Intersecting)
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Dependent	Furmaata baay'ee daangaa hin qabne	Walirra bu'aa (Coincident)
$\frac{a_1}{a_2} = \frac{b_1}{b_2}$ $\frac{c_1}{c_2}$ wajjin wal qixa miti	Inconsistent	Furmaata hin qabu	Cinaattii (Parallel)

Yoom akka itti fayyadamtu

Reeshoo koo-efisiyantii fayyadamuun sirni jijjiiramaa lamaa furmaata qabaachuu fi dhabuu isaa baruuf itti fayyadamaa.

Of eeggadhu

Koo-efisiyantii walbira qabuu dura himoonni lamaan bifa dhaabbataa walfakkaatuun barreeffamuu isaanii mirkaneessaa.

Grade 9 Mathematics, fuula 87-129 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Gaaffii shaakalaa boqonnaa 3

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

Solve the linear equation 3x + 4 = 10 for x.

A.x = 2
B.x = 3
C.x = 6
D.x = 14

Deebii fi sababa agarsiisi

Deebii: A x = 2

Sababa

Subtract 4 from both sides: 3x = 10 - 4 = 6. Then divide both sides by 3 to get x = 2.

Gaaffii 2

What are the solutions to the absolute value equation |x| = 4?

A.x = 4 only
B.x = -4 only
C.x = 4 or x = -4
D.No real solution

Deebii fi sababa agarsiisi

Deebii: C x = 4 or x = -4

Sababa

By the definition of absolute value, for any non-negative number p, |x| = p means x = p or x = -p. Therefore, x = 4 or x = -4.

Gaaffii 3

Solve the exponential equation 2^x+4 = 32.

A.x = 1
B.x = 2
C.x = 5
D.x = 9

Deebii fi sababa agarsiisi

Deebii: A x = 1

Sababa

Write 32 with base 2: 32 = 2⁵. Since bases are equal, equate the exponents: x + 4 = 5, which gives x = 1.

Gaaffii 4

For a quadratic equation with real coefficients, what is true about the roots if the discriminant D = b² - 4ac > 0?

A.There are no real roots
B.The roots are real and distinct
C.The roots are real and equal
D.The equation has infinitely many solutions

Deebii fi sababa agarsiisi

Deebii: B The roots are real and distinct

Sababa

When the discriminant D > 0, the quadratic equation has two distinct (unequal) real roots.

Gaaffii 5

What is the solution (x, y) to the following system of linear equations?
x + 2y = 1
-x + y = 5

A.(3, -4)
B.(-3, 2)
C.(2, -3)
D.(-1, 1)

Deebii fi sababa agarsiisi

Deebii: B (-3, 2)

Sababa

Add the two equations together: (x + -x) + (2y + y) = 1 + 5, which gives 3y = 6, so y = 2. Substituting into -x + 2 = 5 gives x = -3. Thus, the solution is (-3, 2).

Gaaffii 6

What are the solutions to the equation |x + 7| = 14?

A.x = 7 or x = -21
B.x = 7 or x = -7
C.x = 14 or x = -14
D.x = 21 or x = -7

Deebii fi sababa agarsiisi

Deebii: A x = 7 or x = -21

Sababa

Set up two linear equations: x + 7 = 14 or x + 7 = -14. Solving these gives x = 14 - 7 = 7 or x = -14 - 7 = -21.

Gaaffii 7

Solve the quadratic equation x² - 6x - 16 = 0 by factorization.

A.x = -8 or x = 2
B.x = 8 or x = -2
C.x = 4 or x = -4
D.x = -16 or x = 1

Deebii fi sababa agarsiisi

Deebii: B x = 8 or x = -2

Sababa

Find two numbers whose sum is -6 and product is -16. These are -8 and 2. Factorizing gives (x - 8)(x + 2) = 0, so x = 8 or x = -2.

Gaaffii 8

What is the solution to the radical equation √(15 - 2x) = x?

A.x = 3 and x = -5
B.x = -5 only
C.x = 3 only
D.No real solution

Deebii fi sababa agarsiisi

Deebii: C x = 3 only

Sababa

Squaring both sides gives 15 - 2x = x², or x² + 2x - 15 = 0, which factors into (x - 3)(x + 5) = 0. Since x ≥ 0 for the square root, x = -5 cannot be a solution. The only solution is x = 3.

Gaaffii 9

Tickets to a circus in Addis Ababa cost Birr 25 for children and Birr 50 for adults. On one day, total attendance was 2,000 people and total revenue was Birr 70,000. How many children bought tickets?

A.800
B.1,000
C.1,200
D.1,500

Deebii fi sababa agarsiisi

Deebii: C 1,200

Sababa

Let c be children and a be adults. The system is c + a = 2000 and 25c + 50a = 70000. Substituting a = 2000 - c into the second equation gives 25c + 50(2000 - c) = 70000, which simplifies to -25c = -30000, so c = 1200.

Gaaffii 10

If the difference between the roots of the quadratic equation x² - 13x + k = 0 is 7, what is the value of k?

A.30
B.20
C.13
D.40

Deebii fi sababa agarsiisi

Deebii: A 30

Sababa

Let the roots be r₁ and r₂. From the coefficients, r₁ + r₂ = 13. Given r₁ - r₂ = 7, solving the two equations gives r₁ = 10 and r₂ = 3. Therefore, k = r₁ × r₂ = 10 × 3 = 30.

Appiin boqonnaa kana irratti gaaffii 12 qaba; utuu hojjettuu ni sirreessa. Kan asii jiran 10 jalqabaa ti.