# How do you solve x^2+2x-5=0?

Mar 28, 2016

$- 1 \pm \sqrt{6}$

#### Explanation:

$D = {d}^{2} = {b}^{2} - 4 a c = 4 + 20 = 24$ --> $d = \pm 2 \sqrt{6}$
There are 2 real roots:
$x = - \frac{2}{2} \pm \frac{2 \sqrt{6}}{2} = - 1 \pm \sqrt{6}$

Mar 28, 2016

$x = - 1 \pm \sqrt{6}$

#### Explanation:

color(blue)(x^2+2x-5=0

This is a Quadratic equation (in form $a {x}^{2} + b x + c = 0$)

color(brown)(x=(-b+-sqrt(b^2-4ac))/(2a)

Where

color(red)(a=1,b=2,c=-5

$\rightarrow x = \frac{- 2 \pm \sqrt{{2}^{2} - 4 \left(1\right) \left(- 5\right)}}{2 \left(1\right)}$

$\rightarrow x = \frac{- 2 \pm \sqrt{4 - 4 \left(- 5\right)}}{2}$

$\rightarrow x = \frac{- 2 \pm \sqrt{4 - \left(- 20\right)}}{2}$

$\rightarrow x = \frac{- 2 \pm \sqrt{4 - \left(- 20\right)}}{2}$

$\rightarrow x = \frac{- 2 \pm \sqrt{4 + 20}}{2}$

$\rightarrow x = \frac{- 2 \pm \sqrt{24}}{2}$

$\rightarrow x = \frac{- 2 \pm \sqrt{4 \cdot 6}}{2}$

$\rightarrow x = \frac{- 2 \pm 2 \sqrt{6}}{2}$

$\rightarrow x = \frac{- \cancel{2} \pm \cancel{2} \sqrt{6}}{\cancel{2}}$

color(green)(rArrx=-1+-sqrt6

Mar 28, 2016

$\textcolor{b l u e}{\implies x \approx 1.449 \text{ or "x~~-3.449" to 3 decimal places}}$

#### Explanation:

Another approach would be to complete the square. It is another form of the standard $x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$ .

$\textcolor{b r o w n}{\text{It has to be another form otherwise you would not be}}$$\textcolor{b r o w n}{\text{able to solve for x when y=0.}}$

Given:$\text{ } {x}^{2} + 2 x - 5 = 0$

Note that standard form of $\text{ } y = a {x}^{2} + b x + c$ becomes

$y = a {\left(x + \frac{b}{2 a}\right)}^{2} + c + k \text{ }$ where $k$ is a corrective constant

$\implies y = {\left(x + 1\right)}^{2} - 5 + k$

The error comes from ${\left(\frac{b}{2}\right)}^{2} \to {\left(+ \frac{2}{2}\right)}^{2}$

So ${\left(+ \frac{2}{2}\right)}^{2} + k = 0 \text{ } \implies k = - 1$ giving

$y = {\left(x + 1\right)}^{2} - 6$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$y + 6 = {\left(x + 1\right)}^{2}$

Taking the square root of both sides

$\sqrt{y + 6} = x + 1$

But for this question $y = 0$ giving

$\pm \sqrt{6} - 1 = x$

$\textcolor{b l u e}{\implies x \approx 1.449 \text{ or "x~~-3.449" to 3 decimal places}}$ 