If for z = α + iβ, |z + 2| = z + 4(1 + i), then α + β and αβ are the roots of the equation

Q: Why can we just equate the imaginary parts to zero? Isn't the equation a complex equation?

Great question! Remember that a complex number is equal to another complex number if and only if their real parts are equal AND their imaginary parts are equal. Since the left side $|z+2|$ is a modulus, it is by definition a real number (the distance from $-2$ to $z$). Therefore, its imaginary part is $0$. This forces the imaginary part of the right side to be $0$ as well.

Q: What if the modulus was negative? Can we have a negative modulus?

Absolutely not! The modulus of a complex number represents a distance in the complex plane. Distances are always non-negative. If you ever find yourself with a negative modulus, recheck your steps—you might have made a sign error.

Q: Do I need to check for extraneous roots after squaring the equation?

Yes, that is a very sharp observation! When we square both sides of an equation, we can sometimes introduce extraneous roots. In this case, we have $\sqrt{(\alpha+2)^2 + \beta^2} = \alpha + 4$. Since the left side is a square root (which is non-negative), the right side $\alpha + 4$ must also be non-negative. Always check if your final value of $\alpha$ satisfies $\alpha + 4 \ge 0$.

Question

If for $z = \alpha + i\beta$, $|z + 2| = z + 4(1 + i)$, then $\alpha + \beta$ and $\alpha\beta$ are the roots of the equation

Options:

(A) $x^2 + 3x - 4 = 0$

(B) $x^2 + 7x + 12 = 0$

(C) $x^2 + x - 12 = 0$

(D) $x^2 + 2x - 3 = 0$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a complex number $z = \alpha + i\beta$ and the equation $|z + 2| = z + 4(1 + i)$. Our first step is to substitute $z = \alpha + i\beta$ into the equation:

$$|(\alpha + i\beta) + 2| = (\alpha + i\beta) + 4(1 + i)$$

Grouping the real and imaginary parts, we obtain:

$$|(\alpha + 2) + i\beta| = (\alpha + 4) + i(\beta + 4)$$

### The Master Insight

The modulus $|(\alpha + 2) + i\beta|$ represents the distance from the origin to the point $(\alpha+2, \beta)$, which is defined as $\sqrt{(\alpha + 2)^2 + \beta^2}$. Since this modulus is a real number, the right side of the equation must also be purely real.

This implies that the imaginary part of the right side must be zero:

$$\beta + 4 = 0 \Rightarrow \beta = -4$$

### Solving for the Real Part

With $\beta = -4$ determined, we equate the real parts of the equation:

$$\sqrt{(\alpha + 2)^2 + \beta^2} = \alpha + 4$$

Substituting $\beta = -4$ into the expression, we get:

$$\sqrt{(\alpha + 2)^2 + (-4)^2} = \alpha + 4$$

Squaring both sides to eliminate the radical yields:

$$(\alpha + 2)^2 + 16 = (\alpha + 4)^2$$

Expanding the binomials results in:

$$\alpha^2 + 4\alpha + 4 + 16 = \alpha^2 + 8\alpha + 16$$

Canceling $\alpha^2$ from both sides and simplifying the linear terms:

$$4\alpha + 20 = 8\alpha + 16 \Rightarrow 4 = 4\alpha \Rightarrow \alpha = 1$$

Thus, our complex number is $z = 1 - 4i$.

### Final Calculation

We are tasked with forming a quadratic equation whose roots are $R_1 = \alpha + \beta$ and $R_2 = \alpha\beta$. Calculating these values:

$$R_1 = 1 + (-4) = -3$$
$$R_2 = 1 	imes (-4) = -4$$

The quadratic equation is given by $x^2 - (	ext{sum of roots})x + (	ext{product of roots}) = 0$. The sum of these new roots is:

$$-3 + (-4) = -7$$

The product of these new roots is:

$$(-3) 	imes (-4) = 12$$

Substituting these into the standard form, we arrive at the final quadratic equation:

**$$x^2 + 7x + 12 = 0$$**

Animated Solution for Mathematics - Complex Numbers: If for $z = α + i β$ , $∣ z + 2∣ = z + 4 (1 + i)$ , then $α + β$ and $α β$ are the roots of the equation

Select Answer:

Visualized Solution

Defining the Complex Equation

Substitution into the Equation

Grouping Real and Imaginary Parts

Applying the Modulus Formula

Equating Imaginary Parts

Equating Real Parts

Squaring and Expanding

Solving for $α$

Finding the Roots of the New Equation

Constructing the Quadratic Equation

The Sigma Insight: Algebraic Operations on Complex Numbers

Analyzing the Setup

The Master Insight

Solving for the Real Part

Final Calculation

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Animated Solution for Mathematics - Complex Numbers: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If for z=α+iβ, ∣z+2∣=z+4(1+i), then α+β and αβ are the roots of the equation