Question

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

Answer 1

Hint: Use the completing the square method to modify the quadratic equation. Afterwards, while solving for $x$ , count the number of linear equations in real coefficients that we are able to get by using this method. Then use this count to determine the nature of the solutions of the quadratic equation (real or not).

Complete step by step answer:
To use the completing the square method, we divide the quadratic equation by $2$ on both sides to ensure that the coefficient of ${{x}^{2}}$ is $1$ .
${{x}^{2}}-2x+\dfrac{3}{2}=0$
Then we observe that by adding and subtracting the squares of half the coefficient of $x$ , i.e 1, we obtain
${{x}^{2}}-2x+1-1+\dfrac{3}{2}=0$
Observe that the first three terms in the above equation are only the terms in the expansion of $\left(x-1\right)^2$ . Hence we obtain
$\left(x-1\right)^2+\dfrac12=0$
Now, we rewrite this equation and get $\left(x-1\right)^2=-\dfrac12$ . Clearly, it is not possible to take square roots on both sides since the term on the right hand side of this equation is negative and the square root of a negative number does not belong to the set of real numbers.
Therefore, we get no linear equations in real coefficients by using the completing the square method to solve the quadratic equation.

Hence, we conclude that the nature of solutions of this quadratic equation is non-real (or complex).

Note: The nature of the solutions of a quadratic equation of the form $a{{x}^{2}}+bx+c=0$ can be found directly by using the formula for the discriminant of the quadratic equation: $D={{b}^{2}}-4ac$.
If $D\ge 0$ , the solutions are real else if $D<0$ , the solutions are non-real (or complex). For the equation $2{{x}^{2}}-4x+3=0$ , we see that $a=2,\ b=-4,\ c=3$ and $D={{b}^{2}}-4ac=16-4(2)(3)=-8<0$. Hence, the solutions of this equation are non-real (or complex). Observe that it is important to ensure that the equation is in standard form before using the formula for the discriminant.