Question

How do you solve the quadratic using the quadratic formula given \[3{d^2} - 5d + 6 = 0\] over the set of complex numbers?

Answer 1

Hint: A polynomial of degree two is called a quadratic polynomial and its zeros can be found using many methods like factorization, completing the square, graphs, quadratic formula etc. The quadratic formula is used when we fail to find the factors of the equation. In the given question we have to solve the given quadratic equation using the quadratic formula. That is \[d = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] .

Complete step by step solution:
Given, \[3{d^2} - 5d + 6 = 0\] .
We know the standard quadratic equation is of the form \[a{d^2} + bd + c = 0\] . Comparing the given problem we have \[a = 3\] , \[b = - 5\] and \[c = 6\] .
We have the quadratic formula \[d = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\] .
Substituting the values we have,
 \[d = \dfrac{{ - ( - 5) \pm \sqrt {{{( - 5)}^2} - 4(3)(6)} }}{{2(3)}}\]
 \[ = \dfrac{{5 \pm \sqrt {25 - 72} }}{6}\]
 \[ = \dfrac{{5 \pm \sqrt { - 47} }}{6}\]
We know that \[\sqrt { - 1} = i\] ,
 \[ = \dfrac{{5 \pm i\sqrt {47} }}{6}\]
Hence we have the roots,
 \[ \Rightarrow d = \dfrac{{5 + i\sqrt {47} }}{6}\] and \[d = \dfrac{{5 - i\sqrt {47} }}{6}\]
So, the correct answer is “\[ d = \dfrac{{5 + i\sqrt {47} }}{6}\] and \[d = \dfrac{{5 - i\sqrt {47} }}{6}\]”.

Note: In the above we have complex roots. Note that complex numbers always exist in pairs.Since the degree of the given polynomial is 2. Hence we have two roots. In various fields of mathematics require the point at which the value of a polynomial is zero, those values are called the factors/solution/zeros of the given polynomial. On the x-axis, the value of y is zero so the roots of an equation are the points on the x-axis that is the roots are simply the x-intercepts. The quadratic formula is also called Sridhar’s formula. Careful in the calculation part.