If the quadratic equation z2+(a+i b) z+c+i d=0, where a, b, c, d are non-zero real numbers, has a real root, then

Question

If the quadratic equation z2+(a+i b) z+c+i d=0, where a, b, c, d are non-zero real numbers, has a real root, then (A) d2-a b d-c2=0 (B) d2-a b d+b2 c=0 (C) d2+a b d+c2=0 (D) None of these

Tardigrade Admin · Accepted Answer

Let α be a real root of the given equation. Then, α2+(a+i b) α+c+i d=0 ⇒ α2+a α+c=0 and b α+d=0 ⇒ (-(d/b))2+a(-(b/d))+c=0 ⇒ (d2/b2)-(a d/b)+c=0 or d2-a b d+b2 c=0

Q. If the quadratic equation $z^{2} + (a + ib) z + c + i d = 0$ , where $a, b, c, d$ are non-zero real numbers, has a real root, then

$d^{2} - ab d - c^{2} = 0$

$d^{2} - ab d + b^{2} c = 0$

$d^{2} + ab d + c^{2} = 0$

None of these

Let $α$ be a real root of the given equation.
Then, $α^{2} + (a + ib) α + c + i d = 0$
$\Rightarrow α^{2} + a α + c = 0$ and $b α + d = 0$
$\Rightarrow (- \frac{d}{b})^{2} + a (- \frac{b}{d}) + c = 0$
$\Rightarrow \frac{d ^{2}}{b ^{2}} - \frac{a d}{b} + c = 0$
or $d^{2} - ab d + b^{2} c = 0$

Q. If the quadratic equation z2+(a+ib)z+c+id=0, where a,b,c,d are non-zero real numbers, has a real root, then

d2−abd−c2=0

d2−abd+b2c=0

d2+abd+c2=0