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Solve the inequality 3x+10+x^2<0

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Answer

$-1.5-2.7839i<x<-1.5+2.7839i$

Step by step solution

Problem

$x^2+3x+10<0$
1

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where $a=1$, $b=3$ and $c=10$

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

Substituting the values of the coefficients of the equation in the quadratic formula

$x=\frac{3\left(-1\right)\pm \sqrt{3^2-40}}{2}$
3

Multiply $-1$ times $3$

$x=\frac{-3\pm \sqrt{3^2-40}}{2}$
4

Calculate the power

$x=\frac{-3\pm \sqrt{9-40}}{2}$
5

Add the values $9$ and $-40$

$x=\frac{-3\pm \sqrt{-31}}{2}$
6

Calculate the power using complex numbers

$x=\frac{-3\pm \sqrt{30}i}{2}$
7

To obtain the two solutions, divide the equation in two equations, one when $\pm$ is positive ($+$), and another when $\pm$ is negative ($-$)

$x_1=\frac{-3+ \sqrt{30}i}{2}\:\:,\:\:x_2=\frac{-3- \sqrt{30}i}{2}$
8

Simplifying

$x_1=-1.5+2.7839i,\:x_2=-1.5-2.7839i$
9

Applying the quadratic formula we obtained the two solutions $x_1$ and $x_2$, with which we write the solution interval

$-1.5-2.7839i<x<-1.5+2.7839i$

Answer

$-1.5-2.7839i<x<-1.5+2.7839i$

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