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Find the break even points of the expression $x^2-x+1=0$

Step-by-step Solution

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Final Answer

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
Got another answer? Verify it here!

Step-by-step Solution

Specify the solving method

1

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

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

Simplifying

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

Any expression multiplied by $1$ is equal to itself

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

Multiply $-1$ times $-1$

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

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

$x=\frac{1\pm \sqrt{1-4}}{2}$

Add the values $1$ and $-4$

$x=\frac{1\pm \sqrt{-3}}{2}$
2

Simplifying

$x=\frac{1\pm \sqrt{-3}}{2}$
3

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

$x=\frac{1+\sqrt{-3}}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{3}$

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

$\sqrt{3}i$

Calculate the power $\sqrt{3}$

$\sqrt{3}i$
4

Calculate the power $\sqrt{-3}$ using complex numbers

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{3}$

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{-3}}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

$\sqrt{3}i$

Calculate the power $\sqrt{3}$

$\sqrt{3}i$

Calculate the power $\sqrt{3}$

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$

Calculate the power $\sqrt{-3}$ using complex numbers

$-\sqrt{3}i$

Calculate the power $\sqrt{3}$

$-\sqrt{3}i$
5

Calculate the power $\sqrt{-3}$ using complex numbers

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
6

Multiply $-1$ times $\sqrt{3}$

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$
7

Combining all solutions, the $2$ solutions of the equation are

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$

Final Answer

$x=\frac{1+\sqrt{3}i}{2},\:x=\frac{1-\sqrt{3}i}{2}$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Solve for xFind the rootsSolve by factoringSolve by completing the squareSolve by quadratic formula (general formula)Find the discriminant

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Function Plot

Plotting: $x^2-x+1$

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7
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9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Main Topic: Classify algebraic expressions

An algebraic expression can be classified as a monomial, binomial, trinomial or polynomial, depending on the number of terms.

Used Formulas

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