Question

How do you solve $3{x^2} + 10x - 2 = 0$ by completing the square?

Answer 1

Hint: First of all divide the whole equation by 3, then try to write it in the form ${\left( {a + b} \right)^2} + c = 0$, thus we have it in square form. Take c to RHS and take square root and modify to get the answer.

Complete step-by-step solution:
We are given that we are required to $3{x^2} + 10x - 2 = 0$ by completing the square.
Taking 3 common and crossing it off from the left hand side of the above equation, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} + \dfrac{{10}}{3}x - \dfrac{2}{3} = 0$
We can write the above mentioned equation as follows:-
$ \Rightarrow {x^2} + 2 \times \left( {\dfrac{5}{3}} \right) \times x + {\left( {\dfrac{5}{3}} \right)^2} - {\left( {\dfrac{5}{3}} \right)^2} - \dfrac{2}{3} = 0$ …………………..(1)
Since we know that we have a formula given by the following expression with us:-
$ \Rightarrow {\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
Replacing a by $x$ and b by $\dfrac{5}{3}$, we will then obtain the following equation with us:-
$ \Rightarrow {x^2} + 2 \times \left( {\dfrac{5}{3}} \right) \times x + {\left( {\dfrac{5}{3}} \right)^2} = {\left( {x + \dfrac{5}{3}} \right)^2}$
Putting this in equation number 1, we will then get the following equation:-
$ \Rightarrow {\left( {x + \dfrac{5}{3}} \right)^2} - {\left( {\dfrac{5}{3}} \right)^2} - \dfrac{2}{3} = 0$
Simplifying the calculations on the left hand side, we will then obtain the following equation:-
$ \Rightarrow {\left( {x + \dfrac{5}{3}} \right)^2} - \dfrac{{25}}{9} - \dfrac{2}{3} = 0$
Simplifying the calculations on the left hand side further, we will then obtain the following equation:-
$ \Rightarrow {\left( {x + \dfrac{5}{3}} \right)^2} - \dfrac{{25 + 6}}{9} = 0$
Taking the constants on the right hand side, we will get:-
$ \Rightarrow x = - \dfrac{5}{3} \pm \dfrac{{\sqrt {31} }}{3}$
Thus, we have the required roots.

Note: The students must notice that we have an alternate way of factoring the quadratic equation involved in it as well, if not mentioned that we have to solve it by completing the square. The alternate way is as follows:-
The given equation is $3{x^2} + 10x - 2 = 0$.
Using the quadratic formula given by if the equation is given by $a{x^2} + bx + c = 0$, its roots are given by the following equation:-
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Thus, we have the roots of $3{x^2} + 10x - 2 = 0$ given by:
$ \Rightarrow x = \dfrac{{ - 10 \pm \sqrt {{{(10)}^2} - 4 \times 3 \times ( - 2)} }}{{2(3)}}$
Simplifying the calculations in the square root in the numerator of the right hand side, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 10 \pm \sqrt {124} }}{6}$
Simplifying the calculations in the square root in the numerator of the right hand side further, we will then obtain the following equation with us:-
$ \Rightarrow x = \dfrac{{ - 5 \pm \sqrt {31} }}{3}$
Thus, we have the required roots.