The answer is: [B]: "2 (two) real solutions" .
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Explanation:
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Given: -8x² = 8x − 9 ;
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Let us write this equation in "quadratic format"; that is:
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" ax² + bx + c = 0 ; a ≠ 0 " ;
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So, given: -8x² = 8x − 9 ;
Let us subtract "8x"; and add "9"; to EACH SIDE of the equation;
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-8x² − 8x + 9 = 8x − 9 − 8x + 9 ;
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to get: -8x² − 8x + 9 = 0 ;
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Now, let's multiply the ENTIRE equation (both sides); by "-1", to get rid of the "negative" sign ;
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-1 * { -8x² − 8x + 9 = 0} ;
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to get: → 8x² + 8x − 9 = 0 ;
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This equation is in "quadratic format", which is:
" ax² + bx + c = 0 " ; in which: a = 8; b = 8; and c = -9 ;
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This can be solved by using the "quadratic equation formula" :
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x = {-b ± √(b² − 4ac)} / {2a} ;
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First, let us solve for "2a", that is: "2 * a" ; (the "denominator") ;
→ (2a) = (2*a) = (2*8) = 16 .
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Then, let us solve for: "b² − 4ac" ;
→ 8² − (4*8*-9) = 64 − (-288) = 64 + 288 = 352 ;
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Then, let us solve for: " √(b² − 4ac)"
= √(352) = 18.7616630392937182
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Then, " - b " = "-8" .
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x = (-8 ± 18.7616630392937182) / 16 ;
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So, we have TWO (2) solutions:
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1) x = (-8 + 18.7616630392937182) / 16 = 0.6726039399558573875
and:
2) x = (-8 − 18.7616630392937182) / 16 = -1.6726039399558573875 ;
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which are real solutions.
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The answer is: [B]: "2 real solutions" .
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