If 3 + 4i is a root of the equation x² + px + q = 0, then

Question

Question

Answer 1

Concept:

Whenever a complex number is a root of a polynomial with real coefficients, its complex conjugate is also a root of that polynomial.

The complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.

e.g., z = a + ib then conjugate of z is given by z̅ = a - ib

Quadratic equation in the form of roots: x2 – (sum of roots) x + product of roots = 0

(a + b)(a - b) = (a2 - b2)

i2 = -1

Calculation:

Since complex roots exists in conjugate pairs.

∴ if 3 + 4i is one root of the equation x² + px + q = 0 then other root will be:

3 – 4i

Quadratic equation obtained by these roots is:

x² - (3 + 4i + 3 - 4i)x + (3 + 4i)(3 - 4i) = 0

x² - 6x + 9 + 16 = 0

x2 - 6x +25 = 0

Comparing this equation with the x2 + px + q = 0;

p = -6 and q = 25