Question

Solve using the quadratic formula x(x+6)+4=0

Answer 1

Step 1. The equation that we have is:

$x(x+6)+4=0$

Before we apply the quadratic formula to solve the equation, we need to simplify the expression and multiply x by (x+6):

$\begin{gathered} x(x+6)+4=0 \\ \downarrow \\ x^2+6x+4=0 \end{gathered}$

Step 2. The next step is to compare our equation with the general standard form of the quadratic equation:

$ax^2+bx+c=0$

And we find that the values of x, b, and c are:

$\begin{gathered} x^{2}+6x+4=0 \\ \downarrow\downarrow \\ a=1 \\ b=6 \\ c=4 \end{gathered}$

Step 3. We will use to values of a, b, and c in the quadratic formula:

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

Substituting the known values:

$x=\frac{-6\pm\sqrt{6^2-4(1)(4)}}{2(1)}$

Step 4. Solving the operations step by step:

$\begin{gathered} x=\frac{-6\pm\sqrt{36-16}}{2} \\ \downarrow \\ x=\frac{-6\pm\sqrt{20}}{2} \end{gathered}$

We can simplify the square root of 20 as follows:

$\sqrt{20}=\sqrt{4\cdot5}=2\sqrt{5}$

Therefore:

$x=\frac{-6\pm2\sqrt{5}}{2}$

Now we make the division by 2:

$\begin{gathered} x=\frac{-6\pm2\sqrt{5}}{2} \\ \downarrow \\ x=-3\pm\sqrt{5} \end{gathered}$

Step 5. The final step is to use the + and - signs to find our two solutions:

$\begin{gathered} Solution\text{ 1:} \\ x=-3+\sqrt{5} \\ Solution\text{ 2:} \\ x=-3-\sqrt{5} \end{gathered}$

Answer:

$\begin{gathered} x=-3+\sqrt{5} \\ x=-3-\sqrt{5} \end{gathered}$