Question

Find the roots of quadratic equation $2{x^2} - 4x + 3 = 0$ by completing the square.

Answer 1

Hint: The standard form of quadratic equations is $a{x^2} + bx + c = 0$ where the value of $a$ cannot be $0$.
For using the method of completing the square to find the roots of the quadratic equation, it has to follow the following steps:
Step1: Divide the whole equation by $a$ to get ${x^2} + \dfrac{b}{a}x + \dfrac{c}{a} = 0$.
Step2: Add ${\left( {\dfrac{b}{{2a}}} \right)^2}$on both the sides of the equation.
Step3: Use the whole square identity ${(a + b)^2} = {a^2} + 2ab + {b^2}$, to convert the terms ${x^2} + \dfrac{b}{a}x + {\left( {\dfrac{b}{{2a}}} \right)^2}$ into ${\left( {x + \dfrac{b}{a}} \right)^2}$.
So, the quadratic equation will convert into
${\left( {x + \dfrac{b}{a}} \right)^2} + \dfrac{c}{a} = \dfrac{b}{a}$
Now solve the above equation for variable $x$.

Complete answer:
The given quadratic equation is $2{x^2} - 4x + 3 = 0$.
Comparing the given equation with the standard form of the quadratic equation $a{x^2} + bx + c = 0$, we get
$a = 2,b = - 4$ and $c = 3$
To convert the given quadratic equation in the whole square form, divide the whole equation by $a$,
$\dfrac{{(2{x^2} - 4x + 3)}}{2} = 0$
${x^2} - 2x + \dfrac{3}{2} = 0$
Now, add ${\left( {\dfrac{b}{{2a}}} \right)^2}$ to both the side of the equation,
where ${\left( {\dfrac{b}{{2a}}} \right)^2} = {\left( {\dfrac{{ - 4}}{{2 \times 2}}} \right)^2} = {( - 1)^2} = 1$
${x^2} - 2x + \dfrac{3}{2} + 1 = 1$
Use the whole square identity to reduce the above equation,
$({x^2} - 2x + 1) + \dfrac{3}{2} = 1$
${(x - 1)^2} + \dfrac{3}{2} = 1$
Take all the terms outside the square term to other side of the equality.
${(x - 1)^2} = - \dfrac{3}{2} + 1$
${(x - 1)^2} = - \dfrac{3}{2} + \dfrac{2}{2}$
${(x - 1)^2} = - \dfrac{1}{2}$
Take the square root on both sides of the equation,
$\sqrt {{{(x - 1)}^2}} = \pm \sqrt { - 1.5} $
Since the square root of a negative number is imaginary, so we get
$x - 1 = \pm 1.2247i$
Rearranging the terms,
$x = 1 \pm 1.2247i$
${x_1} = 1 + 1.2247i$ and ${x_2} = 1 - 1.2247i$
So, the roots of the quadratic equation $2{x^2} - 4x + 3 = 0$ are $1 + 1.2247i$ and $1 - 1.2247i$ .

Note:
Another method to solve a quadratic equation is using quadratic formula, that is if $a{x^2} + bx + c = 0$ is a given quadratic equation then their roots can be found by using the formula $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$ . If the given equation is not in the form $a{x^2} + bx + c = 0$ then we can simplify it and bring it to this form so that it will be easier to get the values of $a,b,\& c$