Which of the following is a solution of x2 − 2x = –8? negative 1 minus i square root of 28 1 plus i square root of 28 negative 1 minus i square root of 7 1 plus i square root of 7
Question
Answer:
The correct answer choices are:______________________________________________________
[C]: " 1 minus i square root of 7" ; { " x = 1 − i√7 " } ; AND:
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[D]: " 1 plus i square root of 7" ; { " x = 1 + i√7 " } .
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Explanation:
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Given:
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x² − 2x = - 8 ;
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→ Add "8" to EACH SIDE of the equation; to write this equation in "quadratic format" ; that is:
" ax² + bx + c = 0" ; {a ≠ 0} ;
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→ x² − 2x + 8 = -8 + 8 ;
→ x² − 2x + 8 = 0 ;
→ This is equation is in "quadratic format" ;
That is: " ax² + bx + c = 0" ; {a ≠ 0} ;
in which:
"a = 1 " {implied coefficient of "1";
since "any value; multiplied by "1", is equal to that same value"} ;
"b = -2" ;
"c = 8" ;
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The quadratic equation formula is:__________________________________
x = [-b ± √(b²−4ac] / 2a ;
= ? (Let us plug in our known values for "a, b, & c"
x = { [-(-2] ± [√[(-2²) − (4*1*8)] } / {2* 1} ;
x = [2 ± √(-2)] / 2 ;
x = [2 ± √(4 − 32)] / 2 ;
x = [2 ± √(-28)] / 2 ;
Note: Technically, "√-28" does not exist; since there is "no such thing as the square root of a negative number" .
However, in mathematics, we have designated an "imaginary number", symbolized by the lower case letter, "i" ; to mean: " √-1 " ;
So; " √-28 = √-1 * √28 " = i * √28 ; or, " i√28 " .
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→ "√28 = √4 * √7 = 2√7 ;
→ " i√28 = i* 2√7 = 2i *√7 = 2i√7 ;
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→ So; x = [2 ± √(-28)] / 2 ;
can be rewritten as:
x = 2 ± 2i√7 / 2 ;
and this can be simplied; since we are dividing by "2"; all of the THREE "2 digits" cancel out to "1" ;
→ x = 2 ± 2i√7 / 2 ;
x = 1 ± 1i √7 / 1 ;
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→ Rewrite as:
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x = 1 ± i√7 ;
So there are two (2) solutions:
x = 1 + i√7 ; and x = 1 − i√7.
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The correct answer choices are:
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[C]: " 1 minus i square root of 7" ; { " x = 1 − i√7 " } ; AND:
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[D]: " 1 plus i square root of 7" ; { " x = 1 + i√7 " } .
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solved
general
10 months ago
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