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1

Solved example of square of a trinomial

$x\left(\left(1+x\right)\left(x-1\right)+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
2

Solve the product $\left(1+x\right)\left(x-1\right)$

$x\left(x^2-1\left(1^2\right)+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
3

Calculate the power $1^2$

$x\left(x^2-1\cdot 1+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
4

Apply the formula: $-x$, where $x=1$

$x\left(x^2-1+8\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
5

Subtract the values $8$ and $-1$

$x\left(7+x^2\right)+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x\left(x+2\right)-\left(x-3\right)^2$
6

Solve the product $x\left(x+2\right)$

$7x+x\cdot x^2+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x^2+2x-\left(x-3\right)^2$
7

When multiplying exponents with same base you can add the exponents

$7x+x^{3}+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x^2+2x-\left(x-3\right)^2$
8

Expand $\left(x-3\right)^2$

$7x+x^{3}+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+x^2+2x-x^2-\left(-6x+9\right)$
9

Subtracting $x^2$ and $x^2$

$7x+x^{3}+\left(-x^2-x-1\right)^2=0+x\left(1+x\right)^3+2x-\left(-6x+9\right)$
10

$x+0=x$, where $x$ is any expression

$7x+x^{3}+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+2x-\left(-6x+9\right)$
11

Solve the product $-\left(-6x+9\right)$

$7x+x^{3}+\left(-x^2-x-1\right)^2=x\left(1+x\right)^3+2x+6x-9$
12

Grouping terms

$7x+x^{3}+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3-2x-6x=-9$
13

Adding $-2x$ and $7x$

$5x+x^{3}+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3-6x=-9$
14

Adding $-6x$ and $5x$

$-x+x^{3}+\left(-x^2-x-1\right)^2-x\left(1+x\right)^3=-9$
15

Expand the trinomial using the formula $\left(a+b+c\right)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

$-x+x^{3}+\left(-x^2\right)^2+\left(-x\right)^2+{\left(-1\right)}^2+2-1-1x^2x+2-1-1x^2+2-1-1x-x\left(1+x\right)^3=-9$
16

Multiply $-2$ times $-1$

$-x+x^{3}+\left(-x^2\right)^2+\left(-x\right)^2+{\left(-1\right)}^2+2x^2x+2x^2+2x-x\left(1+x\right)^3=-9$
17

Calculate the power ${\left(-1\right)}^2$

$-x+x^{3}+\left(-x^2\right)^2+\left(-x\right)^2+1+2x^2x+2x^2+2x-x\left(1+x\right)^3=-9$
18

The power of a product is equal to the product of it's factors raised to the same power

$-x+x^{3}+1x^{4}+1x^2+1+2x^2x+2x^2+2x-x\left(1+x\right)^3=-9$
19

Adding $-1x$ and $2x$

$x+3x^2+x^{3}+x^{4}+1+2x^2x-x\left(1+x\right)^3=-9$
20

Rewrite the equation

$3x^2+x+9=0$
21

To find the roots of a polynomial of the form $ax^2+bx+c$ we use the quadratic formula, where in this case $a=3$, $b=1$ and $c=9$. Then substitute the values of the coefficients of the equation in the quadratic formula:<ul><li>$\displaystyle x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$</li></ul>

$x=\frac{-1\pm \sqrt{1-4\cdot 3\cdot 9}}{2\cdot 3}$
22

Multiply $2$ times $3$

$x=\frac{-1\pm \sqrt{1-108}}{6}$
23

Add the values $1$ and $-108$

$x=\frac{-1\pm \sqrt{-107}}{6}$
24

Calculate the power using complex numbers

$x=\frac{-1\pm 10.3441i}{6}$
25

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

$x=\frac{-1+10.3441i}{6},\:x=\frac{-1-10.3441i}{6}$
26

Split the fraction $\frac{-1-10.3441i}{6}$ in two terms with common denominator $6$

$x=-0.1667+\frac{10.3441i}{6},\:x=-0.1667+\frac{-10.3441i}{6}$
27

Solve the equation ($1$)

$x=-0.1667+\frac{10.3441i}{6}$
28

Solve the equation ($2$)

$x=-0.1667+\frac{-10.3441i}{6}$
29

The $2$ solutions of the equation are

$x=-0.1667+1.724i,\:x=-0.1667-1.724i$

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