If the Roots of the Equation (a² + b²)x² − 2b(a + c)x + (b² + c²) = 0 Are Equal, Then a, b, c Are in

A standard quadratic equations problem, not tied to a specific exam paper.

Problem

If the roots of the equation $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$ are equal, then $a$, $b$, $c$ are in:

Note: the worked solution embedded below is titled with the equation $(a^2-b^2)x^2-2c(a+b)x+(c^2+a^2)=0$. Equal roots there give no AP, GP or HP condition, so the standard form of this question is used here.

Key insight. A quadratic has equal roots exactly when its discriminant is zero — and here, setting the discriminant to zero and simplifying doesn’t just give some messy relationship between $a$, $b$, $c$, it collapses all the way down to a perfect square being zero, which pins down $b^2=ac$ directly. That’s precisely the defining condition for $a$, $b$, $c$ to be in geometric progression.

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Approach

Identify the coefficients $A=a^2+b^2$, $B=-2b(a+c)$, $C=b^2+c^2$. Set the discriminant $B^2-4AC=0$ (the condition for equal roots) and expand fully. Simplify the resulting expression — it should factorise into a perfect square, revealing the relationship between $a$, $b$, $c$.

Solution

Step 1 — Write the discriminant condition

$$B^2-4AC=0$$

$$\left[-2b(a+c)\right]^2-4(a^2+b^2)(b^2+c^2)=0$$

Step 2 — Expand the first term

$$4b^2(a+c)^2 = 4b^2(a^2+2ac+c^2) = 4a^2b^2+8ab^2c+4b^2c^2$$

Step 3 — Expand the second term

$$4(a^2+b^2)(b^2+c^2) = 4a^2b^2+4a^2c^2+4b^4+4b^2c^2$$

Step 4 — Subtract and simplify

$$\left[4a^2b^2+8ab^2c+4b^2c^2\right]-\left[4a^2b^2+4a^2c^2+4b^4+4b^2c^2\right]=0$$

$$8ab^2c-4a^2c^2-4b^4=0$$

$$-4\left(a^2c^2-2ab^2c+b^4\right)=0$$

Step 5 — Recognise the perfect square

$$a^2c^2-2ab^2c+b^4 = (ac-b^2)^2$$

So:

$$-4(ac-b^2)^2=0 \implies ac=b^2$$

Step 6 — Interpret

$b^2=ac$ is exactly the condition for $a$, $b$, $c$ to be in geometric progression (each term is the geometric mean of its neighbours).

Answer

Common mistakes

• Mis-expanding $(a+c)^2$ inside the first term. It’s $a^2+2ac+c^2$, not $a^2+c^2$ — dropping the cross term $2ac$ loses the very term that eventually combines to form the perfect square.
• Losing track of signs while subtracting the two expanded products. Six different terms appear across the two expansions, and several cancel exactly — working through Step 4 carefully term by term (rather than trying to do it in one line) avoids dropping a sign.
• Not recognising $a^2c^2-2ab^2c+b^4$ as $(ac-b^2)^2$. This is the same “perfect square” pattern as $x^2-2xy+y^2=(x-y)^2$, with $x=ac$ and $y=b^2$ — spotting it is what turns a messy quartic expression into the clean condition $b^2=ac$.

Practise next

• If the roots of the equation $(a^2+c^2)x^2-2c(a+b)x+(b^2+c^2)=0$ are equal, then $a$, $c$, $b$ are in what progression?