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-2x2-x-1=0
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User Suleyman
by
8.7k points

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4 votes

Answer:

_______________________________________

→ x = - 1 ± i√7 ;

4

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Or: Write as:

_______________________________________

→ x = -1 + i√7 ; -1 − i√7 ;

4 4

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Explanation:

_______________________________________

Given:

-2x² − x − 1 = 0 ;

Solve for " x " .

________________

First, let us multiply BOTH sides of the equation by "-1" ;

to get rid of that "negative sign" in the coefficient: " -2" ;

→ as follows:

________________

-1 * {-2x² − x − 1 } = 0 *{-1} ;

_________________

Rewrite the value: "x" on the "left-hand side" of the question; as: "1x" ;

since "any value" ; multipled by "1" ; results in that same individual value:

_________________

-1 * {-2x² − 1x − 1 } = 0 *{-1} ;

________________

Note the "distributive property" of multiplication:

________________

a(b + c) = ab + ac ;

________________

Likewise, from the "left-hand side" of the equation:

-1 * {-2x² − 1x − 1 } = {-1 * -2x²} + {-1 * -1x} + {-1* -1 } ;

= 2x² + 1x + 1 ;

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Rewrite the "left-hand side" of the equation:

2x² + 1x + 1 ;

Now, the "right-hand side of the equation:

" { 0 * -1 } = 0 " .

Now, rewrite the entire equation:

2x² + 1x + 1 = 0 ;

________________

Note: This equation is written in "quadratic format" ;

that is: " ax² + bx + c " ; [a ≠ 0] ;

→ in which:

a = 2 ; b = 1 ; c = 1 ;

________________

→ which means we can solve for the value(s) for "x" ; by using the

"quadratic equation formula" ;

→ that is:

________________

x = -b ± √(b² − 4ac) ;

2a

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So; to solve for the values for "x" ; we substitute our known values for:

"a" ; "b" ; and "c" ; and solve:

________________

x = -1 ± √(1² − 4*2 *1) ;

4

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→ x = -1 ± √(1 − 8) ;

4

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→ x = -1 ± √(-7) ;

4

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Note: "√(-7)" ; is the square root of a negative number.

Note that we use the lower case letter, " i " ;

as a symbol to represent the imaginary number: " √(-1) " .

This is an "imaginary number" ; since one cannot take the "square root" of a "number lower than zero."

So, we write: √-7 ; as: " i * 7 " ; or, just: " i √7 " ; since:

"i = √-1" ; and since: " √-7" would "factor out" to: "√7 * √-1 " .

_________________________

Note: In the mainstream U.S.A. , the concept of " i " as the "imaginary number" — " √-1 " ; is usually introduced in the second year college-prep algebra class (after "first year high school algebra" in 9th grade; and after "geometry" in 10th grade).

_________________________

So; we have:

→ x = -1 ± √(-7) ;

4

___________

We can rewrite this as:

____________

→ x = -1 ± i√7 ;

4

_____________

We can leave our answer written like that.

We have two (2) solutions—so we can also write out both solutions for the value of x:

_____________

→ x = -1 + i√7 ; -1 + i√7 ;

4 4

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Hope this explanation is helpful to you!

Best of luck— and best wishes!

_____________________________________________

User Tim Eckel
by
6.8k points

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