Question

Factorise \[ - 3{x^2} + 2x + 2\]?

Answer 1

Hint: In this question we have to factorise the given polynomial using quadratic formula. In the polynomial\[a{x^2} + bx + c\], where "\[a\]", "\[b\]", and “\[c\]" are real numbers and the Quadratic Formula is derived from the process of completing the square, and is formally stated as:
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]

Complete step-by-step solution:
Quadratic equations are the equations that are often called second degree. It means that it consists at least one term which is squared, the general form of quadratic equation is\[a{x^2} + bx + c = 0\], where "\[a\]", "\[b\]", and "\[c\]" are numerical coefficients or constant, and the value of\[x\]is unknown. And one fundamental rule is that the value of\[a\], the first constant cannot be zero in a quadratic equation.
Now the given quadratic equation is,
\[ - 3{x^2} + 2x + 2\],
The given polynomial can be rewritten as,
\[ = - \left( {3{x^2} - 2x - 2} \right)\],
Now using the quadratic formula, which is given by\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\],
Here\[a = 3\],\[b = - 2\],\[c = - 2\],
Now substituting the values in the formula we get,
\[\].,
Now simplifying we get,
\[ \Rightarrow x = \dfrac{{2 \pm \sqrt {4 - \left( { - 24} \right)} }}{6}\],
Now again simplifying we get,
\[ \Rightarrow x = \dfrac{{2 \pm \sqrt {28} }}{6}\],
Now taking the square root we get,
\[ \Rightarrow x = \dfrac{{2 \pm 2\sqrt 7 }}{6}\],
Now taking common term we get,
\[x = \dfrac{{1 \pm \sqrt 7 }}{3}\],
Now we get two values of \[x\] they are \[x = \dfrac{{1 + \sqrt 7 }}{3}\],\[x = \dfrac{{1 - \sqrt 7 }}{3}\].
Now rewriting the polynomial using the values of \[x\], we get,
\[ \Rightarrow - \left( {x - \left( {\dfrac{{1 + \sqrt 7 }}{3}} \right)\left( {x - \left( {\dfrac{{1 - \sqrt 7 }}{3}} \right)} \right)} \right)\],
The factorising terms of the given polynomial is\[ - \left( {x - \left( {\dfrac{{1 + \sqrt 7 }}{3}} \right)\left( {x - \left( {\dfrac{{1 - \sqrt 7 }}{3}} \right)} \right)} \right)\].

\[\therefore \]If we factorise the given equation, i.e.,\[ - 3{x^2} + 2x + 2\], then the polynomial becomes \[ - \left( {x - \left( {\dfrac{{1 + \sqrt 7 }}{3}} \right)\left( {x - \left( {\dfrac{{1 - \sqrt 7 }}{3}} \right)} \right)} \right)\].

Note: Quadratic equation formula is a method of solving quadratic equations, and there are other methods to solve such kinds of solutions, another method used to solve the quadratic equation is by factoring method, in this method we should obtain the solution factorising quadratic equation terms.