How do you solve \[5{x^2} - 2x - 2 = 0\]?

Hint: Here in this question, we have to solve the given equation, 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 solutions.

Complete step-by-step solution:
The question involves the quadratic equation. To the quadratic equation we can find the roots by factoring or by using the formula \[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]. Now we will assume the value of y as zero and consider the given equation. So the equation is written as \[5{x^2} - 2x - 2 = 0\].
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=5 b=-2 and c=-2. Now substituting these values to the formula for obtaining the roots we have
\[x = \dfrac{{ - ( - 2) \pm \sqrt {{{( - 2)}^2} - 4(5)( - 2)} }}{{2(5)}}\]
On simplifying the terms, we have
\[ \Rightarrow x = \dfrac{{2 \pm \sqrt {4 + 40} }}{{10}}\]
Now add 4 and 40 we get
\[ \Rightarrow x = \dfrac{{2 \pm \sqrt {44} }}{{10}}\]
The number 44 is a prime number and we don’t have a square root for this. So, we carry the square root as it is.
Therefore it is written as
\[ \Rightarrow x = \dfrac{{2 \pm 2\sqrt {11} }}{{10}}\]
On simplifying we get
\[ \Rightarrow x = \dfrac{{1 \pm \sqrt {11} }}{5}\]
Therefore, we have \[x = \dfrac{{1 + \sqrt {11} }}{5}\] and \[x = \dfrac{{1 - \sqrt {11} }}{5}\]. We can simplify for further so we get
$x=0.8633$ or $x=-0.4633$

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.