How do you solve using the quadratic formula for $x^{2} + x + 5 = 0$ $x^2 + x + 5 = 0$ ?

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How do you solve using the quadratic formula for $x^{2} + x + 5 = 0$ $x^2 + x + 5 = 0$ ?

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Answer 1 · 2018-04-13T20:13:27

The answer is $\frac{- 1 \pm i \sqrt{19}}{2}$ $(-1+-isqrt(19))/2$ .

Explanation:

The quadratic formula is $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x=(-b+-sqrt(b^2-4ac))/(2a$ for the equation $a x^{2} + b x + c$ $ax^2+bx+c$ .

In this case, $a = 1$ $a=1$ , $b = 1$ $b=1$ , and $c = 5$ $c=5$ .

You can therefore substitute in those values to get:

$\frac{- 1 \pm \sqrt{1^{2} - 4 (1) (5)}}{2 (1)}$ $(-1+-sqrt(1^2-4(1)(5)))/(2(1)$ .

Simplify to get $\frac{- 1 \pm \sqrt{- 19}}{2}$ $(-1+-sqrt(-19))/2$ .

Because $\sqrt{- 19}$ $sqrt(-19)$ is not a real number, we have to stick to imaginary solutions. (If this problem asks for real number solutions, there are none.)

The imaginary number $i$ $i$ equals $\sqrt{- 1}$ $sqrt(-1)$ , therefore we can substitute it in:

$\frac{- 1 \pm \sqrt{- 1 \cdot 19}}{2} \to \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{19}}{2} \to \frac{- 1 \pm i \sqrt{19}}{2}$ $(-1+-sqrt(-1*19))/2 rarr (-1+-sqrt(-1)*sqrt(19))/2 rarr (-1+-isqrt(19))/2$ , the final answer.

Hope this helps!

Answer 2 · 2018-04-13T20:16:58

See application of the quadratic formula below in obtaining the result:
$XXX x = - \frac{1}{2} \pm \sqrt{19} i$ $color(white)("XXX")x=-1/2+-sqrt(19)i$

Explanation:

$x^{2} + x + 5 = 0$ $x^2+x+5=0$ is equivalent to $1 x^{2} + 1 x + 5 = 0$ $color(red)1x^2+color(blue)1x+color(magenta)5=0$

Applying the general quadratic formula $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ $x=(-color(blue)b+-sqrt(color(blue)b^2-4color(red)acolor(magenta)c))/(2color(red)a$
for $a x^{2} + b x + c = 0$ $color(red)ax^2+color(blue)bx+color(magenta)c=0$

to this specific case, we have
$XXX x = \frac{- 1 \pm \sqrt{1^{2} - 4 \cdot 1 \cdot 5}}{2 \cdot 1}$ $color(white)("XXX")x=(-color(blue)1+-sqrt(color(blue)1^2-4 * color(red)1 * color(magenta)5))/(2 *color(red)1)$

$XXXXX = \frac{- 1 \pm \sqrt{- 19}}{2}$ $color(white)("XXXXX")=(-1+-sqrt(-19))/2$

There are no Real solutions, but as Complex values:
$XXX x = - \frac{1}{2} + \sqrt{19} i XXX or XXX x = - \frac{1}{2} - \sqrt{19} i$ $color(white)("XXX")x=-1/2+sqrt(19)icolor(white)("XXX")"or"color(white)("XXX")x=-1/2-sqrt(19)i$

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