Question

Find the real solutions of the quadratic equation $7x^2-2x+4=0$.

Short answer

The given equation has no real solution and its complex solution are $x=\frac{1+3i\sqrt{3}}{7},\frac{1-3i\sqrt{3}}{7}$.

Step-by-step solution

Step 1. By method of quadratic formula.

The roots of the quadratic equation of the form $a{x}^2+bx+c=0$where $a\ne 0$can be evaluated by using the quadratic formula shown below.

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Step 2. Find roots of quadratic equation by quadratic formula.

The given quadratic equation is $7x^2-2x+4=0$whose roots can be evaluated by quadratic formula as shown below.

role="math" localid="1643799236972" $$\begin{gathered} x=\frac{-\left(-2\right)\pm \sqrt{{\left(-2\right)}^2-4\left(7\right)\left(4\right)}}{2\left(7\right)} \\ =\frac{2\pm \sqrt{-108}}{14} \\ =\frac{2\pm 6i\sqrt{3}}{14}\mathrm{....}\left(i=\sqrt{-1}\right) \\ =\frac{1+3i\sqrt{3}}{7},\frac{1-3i\sqrt{3}}{7} \end{gathered}$$

Step 3. Description of steps.

Since the roots of given quadratic equation are complex, i.e. $x=\frac{1+3i\sqrt{3}}{7},\frac{1-3i\sqrt{3}}{7}$. Hence the given equation has no real solution.