This question tests your understanding of complex numbers—numbers in the form $a + bi$ where i is the imaginary unit with $i^2 = -1$—and finding the modulus. Complex numbers extend the real number system to include square roots of negative numbers using $i = \sqrt{-1}$, so $i^2 = -1$. The modulus $|a + bi| = \sqrt{a^2 + b^2}$, like distance from origin. For $3 + 4i$, $|3 + 4i| = \sqrt{9 + 16} = \sqrt{25} = 5$—recognize the 3-4-5 triangle! Choice B correctly computes this. A tempting distractor like Choice C might add without squaring, getting 7. Modulus is always non-negative—keep going, you're doing great!

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = -1—and solving quadratics with complex roots. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. For x² + 2x + 10 = 0, discriminant = 4 - 40 = -36, sqrt(-36) = 6i; solutions x = [-2 ± 6i]/2 = -1 ± 3i. Choice A correctly expresses the roots in a + bi form. A tempting distractor like Choice C might use sqrt(6) instead of simplifying sqrt(36)=6. Complex roots come in conjugate pairs, so this makes sense—fantastic effort solving!

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = $-1$—and powers of i. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of $-1$, so i squared = $-1$. Powers cycle every 4: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. So $i^3 = i^2 \cdot i = -1 \cdot i = -i$—easy pattern! Choice D correctly simplifies to -i. A tempting distractor like Choice C might stop at i without multiplying by i² = -1. Remember the cycle for higher powers too—you're acing this!

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = -1—and how to perform addition while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). For addition, combine real parts together and imaginary parts together: here, z1 + z2 = (5 + 2) + (3 - 7)i = 7 - 4i. Choice A correctly adds the real parts (5 + 2 = 7) and imaginary parts (3 + (-7) = -4), resulting in 7 - 4i. A tempting distractor like Choice C might add the imaginary parts incorrectly by forgetting the negative sign, leading to 7 + 10i instead of subtracting 7. Remember, addition is straightforward: treat real and imaginary parts separately, and you'll get it right every time—great job practicing!

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = -1—and multiplying conjugates. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The product (a + bi)(a - bi) = a² + b², a real number, since the i terms cancel. For (2 + 5i)(2 - 5i) = 2² + 5² = 4 + 25 = 29—simple and always positive! Choice C correctly applies this formula. A tempting distractor like Choice A might forget to replace i squared with -1 or miscalculate, leading to -21. This pattern is super helpful for moduli and division—keep practicing, and it'll become second nature!

This question tests your understanding of complex numbers—numbers in the form a + bi where i is the imaginary unit with i squared = -1—and how to perform multiplication while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. Operations work like algebra with the crucial rule: whenever you see i squared, replace it with -1! For multiplication, use FOIL: (3 + 2i)(1 - 4i) = 31 + 3(-4i) + 2i1 + 2i(-4i) = 3 - 12i + 2i - 8i² = 3 - 10i + 8 (since -8*(-1)=8) = 11 - 10i. Choice A correctly performs the multiplication and simplifies i squared to -1, resulting in 11 - 10i. A tempting distractor like Choice C might forget to replace i squared with -1, leaving it as 3 - 10i - 8i² or mishandling signs. Always simplify i squared right away, and multiplication will be a breeze—keep up the great work!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to plot them on the complex plane. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The complex plane has real parts on the horizontal axis and imaginary on the vertical: z = a + bi corresponds to point (a, b); for 2 - 3i, it's (2, -3). This visualization helps with operations like addition as vector addition. Choice C correctly identifies the coordinates for the point in the complex plane with Re(z) = 2 and Im(z) = -3. Choice A ignores the negative sign in the imaginary part: -3i means y = -3, not +3—pay attention to signs! Plotting aids understanding moduli and arguments: distance from origin is |z|, angle is arg(z)—you're visualizing complex numbers wonderfully, keep practicing!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to solve quadratic equations with complex roots. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. For quadratics $x^2$ + bx + c = 0, solutions are (-b ± $sqrt(b^2$ - 4c))/2; if discriminant is negative, use i: for $x^2$ + 2x + 10 = 0, discriminant 4 - 40 = -36, sqrt(-36) = 6i, so x = (-2 ± 6i)/2 = -1 ± 3i. These solutions are in standard a ± bi form, separating real and imaginary parts. Choice A correctly solves the quadratic and expresses the complex roots in standard form with proper handling of the negative discriminant. Choice C uses sqrt(6)i instead of 3i, perhaps from sqrt(-36) = sqrt(36)*i = 6i, but then dividing by 2 gives 3i, not sqrt(6)i—double-check the arithmetic! Complex roots come in conjugate pairs for real coefficients, and this ensures every quadratic has solutions—amazing job diving into this, you're expanding your math toolkit!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform subtraction while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). For subtraction, combine real parts and imaginary parts after distributing the negative: (6 - 2i) - (1 + 5i) = (6 - 1) + (-2 - 5)i = 5 - 7i. Choice B correctly performs the complex number subtraction and expresses the result in standard form a + bi with accurate sign handling. Choice A forgets to distribute the negative to the imaginary part: subtracting 5i means -5i, not +3i from incorrect combination—always distribute the minus sign! Complex number subtraction summary: (a + bi) - (c + di) = (a - c) + (b - d)i—these operations build your confidence in working with imaginary numbers, keep up the excellent work!

This question tests your understanding of complex numbers—numbers in form a + bi where i is the imaginary unit with i squared = -1—and how to perform addition while keeping results in standard a + bi form. Complex numbers extend the real number system to include square roots of negative numbers using i = square root of -1, so i squared = -1. The standard form a + bi has a as the real part and b as the coefficient of the imaginary part (bi). For addition, combine real parts together and imaginary parts together: (5 + 3i) + (2 - 7i) = (5 + 2) + (3 - 7)i = 7 - 4i. Choice A correctly performs the complex number addition and expresses the result in standard form a + bi with proper combination of like terms. Choice C makes a sign error when combining imaginary parts: adding 3 + 7 instead of 3 - 7, but remember the sign of the imaginary coefficient in z2 is negative! Complex number addition summary: (a + bi) + (c + di) = (a + c) + (b + d)i—it's just like combining like terms in algebra, and you're doing great mastering this foundational skill!