# How do you solve x^2 + x +10 = 0 using the quadratic formula?

Apr 3, 2016

Use the quadratic formula to find roots:

$x = - \frac{1}{2} \pm \frac{\sqrt{39}}{2} i$

#### Explanation:

${x}^{2} + x + 10 = 0$ is of the form $a {x}^{2} + b x + c = 0$ with $a = 1$, $b = 1$ and $c = 10$.

It has discriminant $\Delta$ given by the formula:

$\Delta = {b}^{2} - 4 a c = {1}^{2} - \left(4 \cdot 1 \cdot 10\right) = 1 - 40 = - 39$

Since this is negative, this quadratic equation has no Real roots.

It has a Complex conjugate pair of roots given by the quadratic formula:

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

$= \frac{- b \pm \sqrt{\Delta}}{2 a}$

$= \frac{- 1 \pm \sqrt{- 39}}{2}$

$= \frac{- 1 \pm \sqrt{39} i}{2}$

$= - \frac{1}{2} \pm \frac{\sqrt{39}}{2} i$

Apr 3, 2016

zero

#### Explanation:

${x}^{2} + x + 10 = 0$ is in the form of $a {x}^{2} + b x + c = 0$. Then you apply the quadratic formula

You type into the calculator this:

x=-1+√${1}^{2}$- 4110/2*1

This equals to zero because a surd cannot be a negative number. (Some equations cannot be solved and this is one of them)

Apr 4, 2016

$x = - \frac{1}{2} \pm \frac{\sqrt{39}}{2} i$

#### Explanation:

color(blue)(x^2+x+10=0

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

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

Remember that $a , b \mathmr{and} c$ are the coefficients of ${x}^{2} , x \mathmr{and} 10$

Where,

color(red)(a=1,b=1,c=10

And don't be afraid with the formula!

$\rightarrow x = \frac{- 1 \pm \sqrt{{1}^{2} - 4 \left(1\right) \left(10\right)}}{2 \left(1\right)}$

$\rightarrow x = \frac{- 1 \pm \sqrt{1 - 4 \left(10\right)}}{2}$

$\rightarrow x = \frac{- 1 \pm \sqrt{1 - 40}}{2}$

$\rightarrow x = \frac{- 1 \pm \sqrt{- 39}}{2}$

Oh! we cannot find the square root of $- 39$ because it is a negative number!.Don't worry,in such cases it is called a complex number
(in form $\sqrt{- 1}$) .It is represented as $i$

So,

$\rightarrow x = \frac{- 1 \pm \sqrt{39 \cdot - 1}}{2}$

$\rightarrow x = \frac{- 1 \pm \sqrt{39}}{2} i$

color(green)(rArrx=(-1)/(2)+-(sqrt39)/(2)i