Advertisements
Advertisements
प्रश्न
Solve the following quadratic equation:
\[x^2 - \left( 5 - i \right) x + \left( 18 + i \right) = 0\]
उत्तर
\[ x^2 - \left( 5 - i \right)x + \left( 18 + i \right) = 0\]
\[\text { Comparing the given equation with the general form } a x^2 + bx + c = 0, \text { we get }\]
\[a = 1, b = - \left( 5 - i \right) \text { and } c = \left( 18 + i \right)\]
\[x = \frac{- b \pm \sqrt{b^2 - 4ac}}{2a}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm \sqrt{\left( 5 - i \right)^2 - 4\left( 18 + i \right)}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm \sqrt{\left( 5 - i \right)^2 - 4\left( 18 + i \right)}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm \sqrt{- 48 - 14i}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\sqrt{48 + 14i}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\sqrt{49 - 1 + 2 \times 7 \times i}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\sqrt{\left( 7 + i \right)^2}}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) \pm i\left( 7 + i \right)}{2}\]
\[ \Rightarrow x = \frac{\left( 5 - i \right) + i\left( 7 + i \right)}{2} \text { or }x = \frac{\left( 5 - i \right) - i\left( 7 + i \right)}{2}\]
\[ \Rightarrow x = 2 + 3i, 3 - 4i\]
\[\text { So, the roots of the given quadratic equation are } 2 + 3i \text { and } 3 - 4i .\]
APPEARS IN
संबंधित प्रश्न
Solve the equation x2 + 3 = 0
Solve the equation 2x2 + x + 1 = 0
Solve the equation –x2 + x – 2 = 0
Solve the equation x2 + 3x + 5 = 0
Solve the equation `sqrt2x^2 + x + sqrt2 = 0`
Solve the equation `x^2 + x/sqrt2 + 1 = 0`
For any two complex numbers z1 and z2, prove that Re (z1z2) = Re z1 Re z2 – Imz1 Imz2
Solve the equation 27x2 – 10x + 1 = 0
\[x^2 - 4x + 7 = 0\]
\[x^2 + 2x + 5 = 0\]
\[21 x^2 + 9x + 1 = 0\]
\[17 x^2 + 28x + 12 = 0\]
\[13 x^2 + 7x + 1 = 0\]
\[2 x^2 + x + 1 = 0\]
\[\sqrt{3} x^2 - \sqrt{2}x + 3\sqrt{3} = 0\]
\[\sqrt{2} x^2 + x + \sqrt{2} = 0\]
\[x^2 + \frac{x}{\sqrt{2}} + 1 = 0\]
\[- x^2 + x - 2 = 0\]
Solving the following quadratic equation by factorization method:
\[x^2 + 10ix - 21 = 0\]
Solve the following quadratic equation:
\[x^2 + 4ix - 4 = 0\]
Solve the following quadratic equation:
\[x^2 - x + \left( 1 + i \right) = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 3\sqrt{2} - 2i \right) x - \sqrt{2} i = 0\]
Solve the following quadratic equation:
\[x^2 - \left( \sqrt{2} + i \right) x + \sqrt{2}i = 0\]
If a and b are roots of the equation \[x^2 - px + q = 0\], than write the value of \[\frac{1}{a} + \frac{1}{b}\].
Write roots of the equation \[(a - b) x^2 + (b - c)x + (c - a) = 0\] .
The complete set of values of k, for which the quadratic equation \[x^2 - kx + k + 2 = 0\] has equal roots, consists of
If a, b are the roots of the equation \[x^2 + x + 1 = 0, \text { then } a^2 + b^2 =\]
If α, β are the roots of the equation \[x^2 + px + 1 = 0; \gamma, \delta\] the roots of the equation \[x^2 + qx + 1 = 0, \text { then } (\alpha - \gamma)(\alpha + \delta)(\beta - \gamma)(\beta + \delta) =\]
The number of real solutions of \[\left| 2x - x^2 - 3 \right| = 1\] is
If the roots of \[x^2 - bx + c = 0\] are two consecutive integers, then b2 − 4 c is
The values of k for which the quadratic equation \[k x^2 + 1 = kx + 3x - 11 x^2\] has real and equal roots are
The set of all values of m for which both the roots of the equation \[x^2 - (m + 1)x + m + 4 = 0\] are real and negative, is
The equation of the smallest degree with real coefficients having 1 + i as one of the roots is
Find the value of a such that the sum of the squares of the roots of the equation x2 – (a – 2)x – (a + 1) = 0 is least.
Show that `|(z - 2)/(z - 3)|` = 2 represents a circle. Find its centre and radius.
