Question

How do you factor \[3{x^2} + 5\]?

Answer 1

Hint: Here in this question, we have to find the factors, the given equation is in the form of a quadratic equation. This is a quadratic equation for the variable x. By using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\], we can determine the roots of the equation and factors are given by $ (x – root_1) $ $ (x – root_2) $

Complete step-by-step answer:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factoring or by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. So the equation is written as \[3{x^2} + 5\].
In general, the quadratic equation is represented as \[a{x^2} + bx + c = 0\], when we compare the above equation to the general form of equation the values are as follows. a=3 b=0 and c=5. Now substituting these values to the formula for obtaining the roots we have
\[roots = \dfrac{{ - 0 \pm \sqrt {{0^2} - 4(3)(5)} }}{{2(3)}}\]
On simplifying the terms, we have
\[ \Rightarrow roots = \dfrac{{0 \pm \sqrt {0 - 60} }}{6}\]
Now subtract 0 from 60 we get
\[ \Rightarrow roots = \dfrac{{0 \pm \sqrt { - 60} }}{6}\]
The number 60 is written as \[15 \times 4\]
\[ \Rightarrow roots = \dfrac{{0 \pm \sqrt { - 15 \times 4} }}{6}\]
The number 4 is a perfect square so we can take out from square root we have
\[ \Rightarrow roots = \dfrac{{ \pm 2\sqrt { - 15} }}{6}\]
On simplifying we have
\[ \Rightarrow roots = \dfrac{{ \pm \sqrt {15} i}}{3}\]
The 15 is written as product of 5 and 3
\[ \Rightarrow roots = \pm \dfrac{{i\sqrt 5 \times \sqrt 3 }}{{\sqrt 3 \times \sqrt 3 }}\]
On cancelling we have’
\[ \Rightarrow roots = \pm \dfrac{{i\sqrt 5 }}{{\sqrt 3 }}\]
Therefore, we have \[root_1 = \dfrac{{ + \sqrt 5 i}}{{\sqrt 3 }}\] or \[root_2 = \dfrac{{ - \sqrt 5 i}}{{\sqrt 3 }}\].
The roots for the quadratic equation when we find the roots by using formula is given by $ (x – root_1) $ $ (x – root_2) $
Substituting the roots values we have
\[ \Rightarrow \left( {x - \dfrac{{\sqrt 5 i}}{{\sqrt 3 }}} \right)\left( {x - \left( { - \dfrac{{\sqrt 5 i}}{{\sqrt 3 }}} \right)} \right)\]
On simplifying we have
\[ \Rightarrow \left( {x - \dfrac{{\sqrt 5 i}}{{\sqrt 3 }}} \right)\left( {x + \dfrac{{\sqrt 5 i}}{{\sqrt 3 }}} \right)\]
Hence we have found the factors for the given equation
So, the correct answer is “ $ \left( {x - \dfrac{{\sqrt 5 i}}{{\sqrt 3 }}} \right)\left( {x + \dfrac{{\sqrt 5 i}}{{\sqrt 3 }}} \right) $ ”.

Note: The quadratic equation can be solved by using the factorisation method and we also find the roots by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. While factorising we use sum product rule, the sum product rule is given as the product factors of the number c is equal to the sum of the factors which satisfies the value of b.