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كويز تفاعلي: 17 Questions: Complex Numbers and Quadratic Functions ريفيل
A comprehensive set of questions focusing on operations with complex numbers and the discriminant of quadratic equations. Includes simplifying radical expressions with negative radicands, powers of i, and identifying the nature of roots from both equations and graphs.
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يرجى الانتباه إلى أن المعلم قام بإعداد الأسئلة فقط، ولم يقم بإعداد الإجابات أو الشروحات المرفقة. وقد تم توليد الإجابات باستخدام تقنيات الذكاء الاصطناعي، لذلك قد تتضمن بعض الأخطاء أو عدم الدقة.
للحصول على الإجابات الصحيحة والمضمونة، يُرجى الرجوع إلى المعلم أو المصدر الدراسي المعتمد.
Question 1
DB question no.: 1
Points: 1
Simplify: $\sqrt{-23} \cdot \sqrt{-46}$
Explanation
$\sqrt{-23} = i\sqrt{23}$ and $\sqrt{-46} = i\sqrt{46}$. Their product is $i^2 \sqrt{23 \cdot 46} = -1 \sqrt{23 \cdot 2 \cdot 23} = -23\sqrt{2}$.
Question 2
DB question no.: 2
Points: 1
Simplify: $(3i)(-2i)(5i) = ?$
Explanation
(3 × -2 × 5)(i × i × i) = -30 i3 = -30(-i) = 30i.
Question 3
DB question no.: 3
Points: 1
Simplify: $i^{11} = ?$
Explanation
i11 = (i4)2 × i3 = 1 × -i = -i.
Question 4
DB question no.: 4
Points: 1
Simplify: $(4i)(-6i)^2 = ?$
Explanation
(4i)(36i2) = (4i)(-36) = -144i.
Question 5
DB question no.: 5
Points: 1
Simplify: $(6 - 3i) + (4 - 2i) = ?$
Explanation
Combine real and imaginary parts: (6+4) + (-3-2)i = 10 - 5i.
Question 6
DB question no.: 6
Points: 1
Simplify: $(-11 + 4i) - (1 - 5i) = ?$
Explanation
(-11 - 1) + (4 - (-5))i = -12 + 9i.
Question 7
DB question no.: 7
Points: 1
Simplify: $(5 - 2i)(4 - i) = ?$
Explanation
20 - 5i - 8i + 2i2 = 20 - 13i - 2 = 18 - 13i.
Question 8
DB question no.: 8
Points: 1
How many real roots does x2 - 8x + 16 = 0 have?
Explanation
The discriminant D = b2 - 4ac = (-8)2 - 4(1)(16) = 64 - 64 = 0. A discriminant of 0 means one repeated real root.
Question 9
DB question no.: 9
Points: 1
What is the discriminant of x2 - 11x - 26 = 0?
Explanation
D = (-11)2 - 4(1)(-26) = 121 + 104 = 225.
Question 10
DB question no.: 10
Points: 1
How many real roots does x2 - 11x - 26 = 0 have?
Explanation
Since the discriminant is 225 (positive), there are two distinct real roots.
Question 11
DB question no.: 11
Points: 1
What is the discriminant of 5x2 - 6 = 0?
Explanation
a = 5, b = 0, c = -6. D = 02 - 4(5)(-6) = 120.
Question 12
DB question no.: 12
Points: 1
What type of roots does 5x2 - 6 = 0 have?
Explanation
The discriminant is 120, which is positive but not a perfect square, resulting in two irrational real roots.
Question 13
DB question no.: 13
Points: 1
What is the discriminant of x2 + 8x + 13 = 0?
Explanation
D = 82 - 4(1)(13) = 64 - 52 = 12.
Question 14
DB question no.: 14
Points: 1
What type of roots does x2 + 8x + 13 = 0 have?
Explanation
The discriminant is 12, which is positive but not a perfect square, indicating two irrational real roots.
Question 15
DB question no.: 15
Points: 1
Describe the discriminant of the related equation of the graph shown.
Explanation
The parabola does not intersect or touch the x-axis, which means the quadratic equation has no real roots and the discriminant is negative.
Question 16
DB question no.: 16
Points: 1
Determine the type and number of roots of the graph shown.
Explanation
The vertex of the parabola lies exactly on the x-axis, meaning it touches the axis at one point, which corresponds to one repeated real root.
Question 17
DB question no.: 17
Points: 1
Describe the discriminant of the related equation of the graph shown.
Explanation
The parabola crosses the x-axis at two distinct points, indicating that the equation has two distinct real roots and a positive discriminant.
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