We have so far looked at all the methods to solve quadratic equations. Now let’s consider equations where the solutions don’t make sense.
Consider a simple quadratic equation like X2 + 4 = 0
By the method of taking square rootswe can very well see that the roots are:
X = ±$\sqrt{-4}$
Now from here we cannot solve further as we very well know that square root of negative numbers is not possible. Hence we stop by saying “No Solution”.
But if one is familiar with complex numbers then we will use the imaginary number i = $\sqrt{-1}$ and write:
X = ±$\sqrt{-4}$ = ±$\sqrt{-1}\sqrt{-1}$ = ±2i
Hence we say that the equation has two imaginary roots i.e. X = 2i, -2i.
Whenever we come across square root of negative numbers we introduce the imaginary number i = $\sqrt{-1}$ and say that the equation has imaginary roots.
Let’s solve a few more examples to see how the solutions look like.
Problem on Imaginary Roots
Solve the quadratic equation 2X2 + 4X + 3 = 0
We cannot factorize this quadratic as there are no factors of (2)(3) = 6 that add up to the middle term 4.
We can apply either the method of completing the square or the quadratic formula.
By quadratic formula:
$$X=\frac{-b\pm\sqrt{b^2-4ac }}{2a}$$
Substitute a=2, b=4, c=3.
$x=\frac{-4\pm\sqrt{4^2-4(2)(3) }}{2(2)}$
$x=\frac{-4\pm\sqrt{-8 }}{4}$
$x=\frac{-4\pm\sqrt{8 }\sqrt{-1 }}{4}$
$x=\frac{-4\pm2\sqrt{2 }i}{4}$
$x=-1\pm\frac{\sqrt{2}}{2}i$
Hence the given equation has two complex roots.
Now in the next example we will take a look at the graph of the solutions as well.
Example of Complex Equation
Solve X2 - 2X + 2 = 0
This cannot be factored.
Here a = 1, b = -2, c = 2
By quadratic formula
$x=\frac{-b\pm\sqrt{b^2-4ac }}{2a}$
$x=\frac{2\pm\sqrt{(-2)^2-4(1)(2) }}{2}$
$x=\frac{2\pm\sqrt{-4}}{2}$
$x=\frac{2\pm2i}{2}$
X = 1± i
The graph of this equation is given by
The roots of an equation in its graph are given by the points where it cuts the x-axis (called as x-intercepts).
As we can see the above graph does not cut the x – axis anywhere. This very well coincides with our calculation that the equation has no real roots.
So an equation with imaginary roots will not have any x – intercepts.
Imaginary Roots of a Quadratic Equation
By imaginary roots we mean the roots to be in the form a + bi where a and b are real numbers and b?0.
These roots can be easily found by using the quadratic formula. One can also determine whether an equation has imaginary roots by looking at its graph.
If the graph has no x-intercepts we can conclude that the equation has imaginary roots.
Related Tags
simple quadratic equation , Complex Equation ,quadratic formula