Prove that if a² + ab + b² = 0 then a = 0 and b = 0

[George Jones]

$$0=a^2 + ab + b^2$$

Complete the square for the first two terms on the right side.

Continuing on with this,

$$\begin{align}
0 &= a^2 + ab + b^2 \\
&= \left(a + \frac{b}{2} \right)^2 - \frac{b^2}{4} + b^2 \\
&= \left(a + \frac{b}{2} \right)^2 + \frac{3}{4} b^2 \\
& = x^2 + y^2
\end{align}$$

where $x = \left(a + \frac{b}{2} \right)$ and $y = b \sqrt{3}/2$. Consequently, $x = 0$ and $y = 0$, which gives $b = 0$ and $a = 0$.

 

[CynicusRex]

I've posted this on Reddit and apparently my proof is sloppy, but not wrong:

"I'm explaining it very broadly because it's been too long that I've been struggling with this. This problem can be found in the book Algebra - I.M. Gelfand, A. Shen. Problem 124. I'm currently self-studying math, and never had a formal training or experience with proofs. Also, we can use determinants, or the factorization of a³-b³ to prove the problem, but those I understand. I do not understand why the one below is wrong. Thanks in advance, any help is appreciated and probably much needed.

Prove that if a²+ab+b² = 0, then a = 0 and b = 0.

a²+ab+b² = 0
a²+b² = -ab

Let's assume a and b are not 0. On the left hand side a² and b² are always positive, so their sum is also always positive. On the right hand side there are four possible situations:

• 1. a and b are both positive (they have the same sign)
• 2. a and b are both negative (they have the same sign)
• 3. a is negative, b is positive (opposite signs)
• 4. a is positive, b is negative (opposite signs)

Situation 1 and 2: if a and b have the same sign, the right hand side is negative:
1. a and b are positive: a²+b² = -ab
2. a and b are negative: (-a)²+(-b)² = -(-a)(-b)
-> a²+b² = -ab

Since the left hand side must be positive, the right hand side must be positive as well for the equation to hold. However, the right hand side must be negative in situation 1 and 2 so the equation does not hold. It only holds when a = b = 0.
Because if only a = 0, then 0²+b² = -0*b -> b² = 0 -> b = 0
and if only b = 0, then a²+0² = -a*0 -> a² = 0 -> a = 0

Situation 3 and 4: if a and b have the opposite sign, the right hand side is positive:
3. a is negative, b is positive: (-a)²+b² = -(-a)b
-> a²+b² = ab
4. a is positive, b is negative: (a)²+(-b)² = -a(-b)
-> a²+b² = ab

If the absolute values of a and b are not equal, then a>b or b>a, therefore the absolute value of the right hand side must be lower than the left hand side because respectively a²>ab and b²>ab.
If the absolute values of a and b are equal, then a=b, and a²+b²=ab becomes 2a² = a², in which evidently 2a²>a² when a is not 0.

So, in all four situations we have shown that the equation only holds when a = b = 0."

The reply:

You're being a bit sloppy with notation in (3) and (4). If a and b have the opposite sign then that doesn't mean that a2 + b2 = ab. I understand that on the right-hand side you're using a to mean -a (or b to mean -b) but that's rather poor practice.

Instead you should either use new variables (e.g. If a is negative, let r = -a so that r is positive and then r2 + b2 = rb) or just use absolute values: If a and b have opposite sign then |a|2 + |b|2 = |a||b|.

Other that that, I don't see a problem with the idea of your proof.

 

[Ray Vickson]

Well late to the party but I've been thinking about extending the domain of this question to including all numbers including complex numbers. This is my proof for it and I'd like to know if it is valid.

Given $a^2+ab+b^2=0$ ∀ $a,b ∈ ℂ$
$a^2+b^2=-ab$ ⇒(Realise that for this to be true only either $a,b∈ℝ$ or both $a,b∉ℝ$)$$ab = \begin{cases} ≤0 & \text{if } a,b ∈ ℝ \\ ≥0 & \text{if } a,b ∉ ℝ \end{cases}$$
$(a+b)^2-2ab=-ab$
$(a+b)^2=ab$
Suppose $a,b≠0$
$⇒(a+b)^2/(ab)=1$
Since $(a+b)^2$ and $ab$ have to be of the same sign, it is a contradiction with the case statement above and hence a,b cannot not be 0.

The result is false for complex numbers. Let $\omega \neq 1$ be a (complex) cube root of unity; that is, $\omega^3 = 1$. If $b = \omega a$ we have $$a^2 + a b + b^2 = a^2 ( 1 + \omega + \omega^2) = 0$$
for any $a$, because $1 + \omega + \omega^2 = 0$.

You can verify all this explicitly by taking
$$ \omega = -\frac{1}{2} + i \frac{\sqrt{3}}{2} .$$

 

[eatsleepmath]

It's easy to simplify this to $(a+b)=\sqrt{ab}$. But extending AM-GM to include negative numbers (which is fine since this setup won't invalidate the underlying stipulations) tells us that $(a+b) \geq 2\sqrt{ab}$. Substitution gives $\sqrt{ab} \geq 2\sqrt{ab}$. The only time this is true is the equality case, i.e. $\sqrt{ab}=0$. Thus at least one of $a$ and $b$ is zero. Solving for the other gives $a^2+a(0)+(0)^2=0$ so the other is zero as well.

 

[Mark44]

Prove that if a²+ab+b² = 0, then a = 0 and b = 0.

a²+ab+b² = 0
a²+b² = -ab

Let's assume a and b are not 0. On the left hand side a² and b² are always positive, so their sum is also always positive. On the right hand side there are four possible situations:

• 1. a and b are both positive (they have the same sign)
• 2. a and b are both negative (they have the same sign)
• 3. a is negative, b is positive (opposite signs)
• 4. a is positive, b is negative (opposite signs)

Situation 1 and 2: if a and b have the same sign, the right hand side is negative:
1. a and b are positive: a²+b² = -ab
2. a and b are negative: (-a)²+(-b)² = -(-a)(-b)
-> a²+b² = -ab

Notwithstanding what someone on Reddit has to say, this is not a good proof. For one thing, it looks like you are concluding that $a^2 + b^2 = -ab$. You should not conclude this -- you are assuming it is true. Since you are assuming that $a^2 + ab + b^2 = 0$, it follows trivially that $a^2 + b^2 = -ab$, regardless of whether a and b are positive, negative, of mixed signs, whatever. The two equations are equivalent. You don't need any machinery to get from one to the other, apart from adding equal quantities to both sides of either equation.

If a and b are both positive or both negative, then the equation $a^2 + b^2 = -ab$ can't possibly be true, and is therefore a contradiction. The left side is positive and the right side is negative. I mentioned this in my earlier post.

The worst part of your proof is that you seem to think putting a minus sign in front of a variable makes it negative. That is NOT true in general. For example, if x = -2, then -x is positive.

Since the left hand side must be positive, the right hand side must be positive as well for the equation to hold. However, the right hand side must be negative in situation 1 and 2 so the equation does not hold. It only holds when a = b = 0.
Because if only a = 0, then 0²+b² = -0*b -> b² = 0 -> b = 0
and if only b = 0, then a²+0² = -a*0 -> a² = 0 -> a = 0