Respuesta :
Answer:
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→ x = - 1 ± i√7 ;
4
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Or: Write as:
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→ x = -1 + i√7 ; -1 − i√7 ;
4 4
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Step-by-step explanation:
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Given:
-2x² − x − 1 = 0 ;
Solve for " x " .
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First, let us multiply BOTH sides of the equation by "-1" ;
to get rid of that "negative sign" in the coefficient: " -2" ;
→ as follows:
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-1 * {-2x² − x − 1 } = 0 *{-1} ;
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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:
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-1 * {-2x² − 1x − 1 } = 0 *{-1} ;
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Note the "distributive property" of multiplication:
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a(b + c) = ab + ac ;
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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 ;
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Note: This equation is written in "quadratic format" ;
that is: " ax² + bx + c " ; [a ≠ 0] ;
→ in which:
a = 2 ; b = 1 ; c = 1 ;
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→ which means we can solve for the value(s) for "x" ; by using the
"quadratic equation formula" ;
→ that is:
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→ 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:
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→ 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 " .
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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).
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So; we have:
→ x = -1 ± √(-7) ;
4
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We can rewrite this as:
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→ x = -1 ± i√7 ;
4
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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:
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→ 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!
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