Question

Solve the given equation.

$x+7x^{\frac{1}{2}}+6=0$

Short answer

There is no real solution.

Step-by-step solution

Step 1. Given information.

The given equation is$x+7x^{\frac{1}{2}}+6=0$.

Step 2. Simplify the equation.

$\begin{array}{rcl}x+7x^{\frac{1}{2}}+6& =& 0\\ {\left(x^{\frac{1}{2}}\right)}^2+7x^{\frac{1}{2}}+6& =& 0\\ {\left(x^{\frac{1}{2}}\right)}^2+6x^{\frac{1}{2}}+x^{\frac{1}{2}}+6& =& 0\\ x^{\frac{1}{2}}\left(x^{\frac{1}{2}}+6\right)+1\left(x^{\frac{1}{2}}+6\right)& =& 0\\ \left(x^{\frac{1}{2}}+6\right)\left(x^{\frac{1}{2}}+1\right)& =& 0\end{array}$

Step 3. Solve the equation for x.

$x^{\frac{1}{2}}=-1$and $x^{\frac{1}{2}}=-6$

The square of a number can't be negative.

Thus, the equation has no real solutions.