Algebra

Complex Numbers

Complex Numbers Problem Solving

         

Let z=a25a36+(a2+6a+8)i z = a^2 - 5a - 36 + (a^2 + 6a + 8)i . What is the sum of all real numbers aa such that z2 z^2 is a negative real number?

Details and assumptions

ii is the imaginary number that satisfies i2=1i^2 = -1.

If the quadratic equation (2+i)x2+(ai)x+462i=0 (2 + i)x^2 + (a - i)x + 46 - 2i = 0 has real roots, what is the value of the positive real constant aa?

Details and assumptions

ii is the imaginary number that satisfies i2=1i^2 = -1.

Let z=x(1i)+8(2+i),z=x(1-i)+8(-2+i), where xx is a real number. If z2z^2 is a negative real number, what is the value of x?x?

Real numbers xx and yy satisfy the equation: x1+i+y1i=14812+2i. \frac{x}{1+i} + \frac{y}{1-i} = \frac{148}{12 + 2i}. What is the value of xy?xy?

Details and assumptions

ii is the imaginary number satisfying i2=1i^2 = -1.

12912+9+12+9129 \frac{12 - \sqrt{-9}}{12 + \sqrt{-9}} + \frac{12 + \sqrt{-9}}{12 - \sqrt{-9}} can be simplified to ab \frac{a}{b} , where aa and bb are coprime positive integers. What is the value of a+b a + b ?

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