Quiz: Complex Quadratic Equations — 8 questions

1. For $a=\alpha+i\beta$ and $z=x+iy$, which system is equivalent to $z^2=a$?

A. $x^2-y^2=\alpha$, $x^2+y^2=\alpha^2+\beta^2$, and $2xy=-\beta$

B. $x^2+y^2=\alpha$, $x^2-y^2=\sqrt{\alpha^2+\beta^2}$, and $2xy=\beta$

C. $x^2-y^2=\alpha$, $x^2+y^2=\sqrt{\alpha^2+\beta^2}$, and $2xy=\beta$

D. $x^2+y^2=\alpha$, $x^2-y^2=\beta$, and $2xy=\sqrt{\alpha^2+\beta^2}$

Answer: (C)

Explanation

Expanding $(x+iy)^2$ and matching real and imaginary parts gives the first and third equations, while adding the squared components yields the modulus relation in the first choice. The second choice swaps the roles of the sum and difference, so it does not represent the required system.

2. What is the primary definition of the square root of a complex number $a=\alpha+i\beta$?

A. It is a complex number that, when squared, results in a purely imaginary number.

B. It is the number that, when multiplied by itself, gives the magnitude of $a$, ignoring the phase or argument.

C. It is a real number that, when squared, equals $a$, regardless of whether $a$ is real or complex.

D. It is a complex number $z=x+iy$ such that $z^2=a$, satisfying the equations $x^2 - y^2 = \alpha$, $x^2 + y^2 = \sqrt{\alpha^2 + \beta^2}$, and $2xy = \beta$.

Answer: (D)

Explanation

The square root of a complex number $a=\alpha+i\beta$ is defined as a complex number $z=x+iy$ that satisfies $z^2=a$, which leads to the equations $x^2 - y^2 = \alpha$, $x^2 + y^2 = \sqrt{\alpha^2 + \beta^2}$, and $2xy = \beta$. This set of equations characterizes the solutions for the square roots in the complex plane.

3. When solving $z^2=\alpha+i\beta$ by writing $z=x+iy$, which procedure correctly determines the possible square roots?

A. Add and subtract the component equations to find $x^2$ and $y^2$, then choose signs satisfying $2xy=\beta$.

B. Solve the real equation for $x$, set $y=\beta$, and use the imaginary equation as a check.

C. Find $x$ and $y$ from the modulus equation, then assign signs without checking their product.

D. Equate $x^2+y^2$ with the real part and determine the imaginary part from the resulting value of $x-y$.

Answer: (A)

Explanation

The standard method first obtains the squares of the real and imaginary components by adding and subtracting the equations, then uses the sign condition $2xy=\beta$. Assigning signs without that condition can produce pairs whose imaginary component has the wrong sign.

4. What is the key equation used to find the square roots of a complex number $a=\alpha+i\beta$?

A. The equations $x^2 + y^2 = \alpha$ and $x^2 - y^2 = \sqrt{\alpha^2 + \beta^2}$ must be satisfied.

B. The equations $x^2 - y^2 = \alpha$, $x^2 + y^2 = \sqrt{\alpha^2 + \beta^2}$, and $2xy = \beta$ must be satisfied.

C. The equations $x^2 + y^2 = \alpha^2 + \beta^2$ and $x^2 - y^2 = \alpha - \beta$ must be satisfied.

D. The equations $x^2 + y^2 = \alpha$ and $x^2 - y^2 = \beta$ must be satisfied.

Answer: (B)

Explanation

The key equations for finding the square roots of a complex number involve expressing the real and imaginary parts as $x^2 - y^2 = \alpha$ and $2xy = \beta$, along with $x^2 + y^2 = \sqrt{\alpha^2 + \beta^2}$. The other options do not correctly represent the system derived from equating $z^2$ to $a$.

5. Which pair gives both solutions of $z^2=3+4i$?

A. $2-i$ and $-2+i$

B. $3+4i$ and $-3-4i$

C. $2+i$ and $-2-i$

D. $\sqrt3+2i$ and $-\sqrt3-2i$

Answer: (C)

Explanation

Squaring $2+i$ gives $3+4i$, and the second square root is its opposite, $-2-i$. The conjugate $2-i$ instead squares to $3-4i$, so it has the wrong imaginary part.

6. What is the main purpose of the complex quadratic formula in solving equations of the form $az^2 + bz + c = 0$?

A. To find the roots of the quadratic equation by calculating the square root of the discriminant and applying it to the formula.

B. To convert the quadratic equation into a linear equation for easier solving.

C. To identify whether the roots of the quadratic are real or complex without calculating them.

D. To determine the coefficients $a$, $b$, and $c$ from the roots of the quadratic equation.

Answer: (A)

Explanation

The complex quadratic formula is used to find the roots of the quadratic equation by calculating the square root of the discriminant, which may be complex, and then applying it to the formula. It does not determine coefficients or convert the equation into a linear form, nor does it directly identify the nature of roots without calculation.

7. What are the two solutions of $z^2=-6-8i$?

A. $-6-8i$ and $6+8i$

B. $2\sqrt2-\sqrt2i$ and $-2\sqrt2+\sqrt2i$

C. $\sqrt2+2\sqrt2i$ and $-\sqrt2-2\sqrt2i$

D. $\sqrt2-2\sqrt2i$ and $-\sqrt2+2\sqrt2i$

Answer: (D)

Explanation

For this equation, the component relations give $x^2=2$ and $y^2=8$, and the negative imaginary part requires $x$ and $y$ to have opposite signs. The third choice uses the same magnitudes but produces a positive imaginary component when the root with positive real part is squared.

8. When was the formula for the roots of a complex quadratic equation, involving the discriminant, first established in mathematical history?

A. It was first introduced in the 17th century by René Descartes.

B. The roots formula was formalized in the early 20th century with the advent of modern algebra.

C. The formula was established in the 19th century during the development of complex analysis.

D. It was known since ancient Greece, with Euclid's work on quadratic equations.

Answer: (C)

Explanation

The complex quadratic formula, involving the discriminant, was formalized in the 19th century as part of the development of complex analysis and algebra. Earlier mathematicians like Descartes contributed to quadratic solutions, but the explicit complex form was established later.