Advertisements
Advertisements
Question
\[x^2 - 4x + 7 = 0\]
Solution
We have:
\[x^2 - 4x + 7 = 0\]
\[ \Rightarrow x^2 - 4x + 4 + 3 = 0\]
\[ \Rightarrow x^2 - 2 \times x \times 2 + 2^2 - (\sqrt{3}i )^2 = 0\]
\[ \Rightarrow (x - 2 )^2 - (\sqrt{3}i )^2 = 0\]
\[ \Rightarrow (x - 2 + \sqrt{3}i) (x - 2 - \sqrt{3}i) = 0\]
\[\Rightarrow (x - 2 + \sqrt{3}i) = 0\] or, \[(x - 2 - \sqrt{3}i) = 0\]
\[\Rightarrow x = 2 - \sqrt{3}i\] or, \[x = 2 + \sqrt{3}i\]
Hence, the roots of the equation are \[2 \pm i\sqrt{3}\] .
APPEARS IN
RELATED QUESTIONS
Solve the equation x2 + 3 = 0
Solve the equation –x2 + x – 2 = 0
Solve the equation `sqrt3 x^2 - sqrt2x + 3sqrt3 = 0`
For any two complex numbers z1 and z2, prove that Re (z1z2) = Re z1 Re z2 – Imz1 Imz2
Solve the equation `x^2 -2x + 3/2 = 0`
Solve the equation 21x2 – 28x + 10 = 0
If z1 = 2 – i, z2 = 1 + i, find `|(z_1 + z_2 + 1)/(z_1 - z_2 + 1)|`
\[x^2 - x + 1 = 0\]
\[x^2 + x + 1 = 0\]
\[17 x^2 + 28x + 12 = 0\]
\[13 x^2 + 7x + 1 = 0\]
\[x^2 + x + \frac{1}{\sqrt{2}} = 0\]
\[\sqrt{5} x^2 + x + \sqrt{5} = 0\]
\[- x^2 + x - 2 = 0\]
Solving the following quadratic equation by factorization method:
\[x^2 + 10ix - 21 = 0\]
Solving the following quadratic equation by factorization method:
\[x^2 + \left( 1 - 2i \right) x - 2i = 0\]
Solving the following quadratic equation by factorization method:
\[6 x^2 - 17ix - 12 = 0\]
Solve the following quadratic equation:
\[x^2 - \left( 2 + i \right) x - \left( 1 - 7i \right) = 0\]
Solve the following quadratic equation:
\[i x^2 - 4 x - 4i = 0\]
Solve the following quadratic equation:
\[x^2 - x + \left( 1 + i \right) = 0\]
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 \[\frac{1}{a} + \frac{1}{b}\].
If \[2 + \sqrt{3}\] is root of the equation \[x^2 + px + q = 0\] than write the values of p and q.
If the difference between the roots of the equation \[x^2 + ax + 8 = 0\] is 2, write the values of a.
If a and b are roots of the equation \[x^2 - x + 1 = 0\], then write the value of a2 + b2.
If α, β are roots of the equation \[x^2 + lx + m = 0\] , write an equation whose roots are \[- \frac{1}{\alpha}\text { and } - \frac{1}{\beta}\].
The number of real roots of the equation \[( x^2 + 2x )^2 - (x + 1 )^2 - 55 = 0\] is
If α, β are the roots of the equation \[x^2 + px + 1 = 0; \gamma, \delta\] the roots of the equation \[x^2 + qx + 1 = 0, \text { then } (\alpha - \gamma)(\alpha + \delta)(\beta - \gamma)(\beta + \delta) =\]
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
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.
If 1 – i, is a root of the equation x2 + ax + b = 0, where a, b ∈ R, then find the values of a and b.
Show that `|(z - 2)/(z - 3)|` = 2 represents a circle. Find its centre and radius.
