# How do you solve x^2+6x+2 = 0 ?

Jan 20, 2018

$x = - 3 + \sqrt{7} \approx - 0.3542$ and $x = - 3 - \sqrt{7} \approx - 5.5457$

#### Explanation:

Solving this by factoring is very difficult because this polynomial doesn't factor super easily.

If you use graphing, you can see the zeros (also called roots) of the euqation that touch the $x$-axis.

graph{x^2+6x+2[-6,1,-8,4]}

However, using the interactive graph, you can see the zeros are not "nice" or "neat", but are rather longer decimal answers.

Next, is completing the square. Here is how you complete the square:

${x}^{2} + 6 x + 2 = 0$

Recall that $y = a {x}^{2} + b x + c$ where $a = 1$, $b = 6$, and $c = 2$.

2) Take the $b$-value (in this case, the 6) and first cut it in half, and then square the result

$\frac{6}{2} = 3 \implies {3}^{2} = 9$

3) Add and subtract this new value, 9, to the equation

${x}^{2} + 6 x + \textcolor{red}{9 - 9} + 2 = 0$

Rearranging with parenthesis, we get

$\left({x}^{2} + 6 x + 9\right) - 9 + 2 = 0$

The part inside the parenthesis is a perfect square

4) Factor the part inside the parenthesis

${\left(x + 3\right)}^{2} - 7 = 0$

5) Finally, solve for $x$

${\left(x + 3\right)}^{2} = 7$

$x + 3 = \pm \sqrt{7}$

$x = - 3 \pm \sqrt{7}$

This final answer are the two roots you saw in the graph above:

$x = - 3 + \sqrt{7} \approx - 0.3542$ and $x = - 3 - \sqrt{7} \approx - 5.5457$

Jan 20, 2018

See below for two of the ways mentioned (I didn't understand if factoring meant some way of taking common factors, or using the quadratic formula).

#### Explanation:

To use a graph to your advantage, you'll need to have it presented to you, such as with a graphing calculator, as you can't really draw it without knowing the solutions to the equations already. Unless you approximate it using calculus, which is probably not the point here. So let's graph $y = {x}^{2} + 6 x + 2$:

graph{x^2+6x+2 [-10, 10, -5, 5]}

it's not visible here, but the points at which the graph intersects the $x$ axis (have a $y$ coordinate of $0$) will be the roots of the equation. Try clicking on those points too on the graph shown.

Now for completing the square. There's an identity that tells us that

${a}^{2} + 2 a b + {b}^{2} = {\left(a + b\right)}^{2}$.

We need to convert the left part of our equation to be of the form seen on the left of the equality above. So, let's add $7$ to both sides:

${x}^{2} + 6 x + 9 = 7$

And now we have $a = x , b = 3$. Therefore:

${\left(x + 3\right)}^{2} = 7$, so

$x + 3 = \pm \sqrt{7} \implies x = \pm \sqrt{7} - 3$

the two solutions.