Answer: The numbers are: " 5, 7, and 9 " .
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Step-by-step explanation:
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Let the "3 consecutive odd numbers" be represented by:
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"x" ;
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"x + 2 " ;
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"x + 4 " ;
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→ Solve for "x" ; and then solve for "(x + 2)" and "(x + 4)" .
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The product of the smaller two numbers, is "46 less than" (the square of) the largest number:
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→ " x* (x + 2) = (x + 4)² − 46 " ;
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Note the "distributive property" of multiplication:
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a(b + c) = ab + ac ;
a(b − c) = ab − ac .
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Start with simplifying the "left-hand side" of the equation:
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" x(x + 2)" = (x * x) + (x * 2 ) = (x² + 2x) ;
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Then simplify the "right-hand side" of the equation:
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" (x + 4)² − 46 " ;
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→ (x + 4)² = (x + 4) (x + 4) ;
Note: " (a + b)(c + d) = ac + ad + bc + bd " .
→ (x + 4)² = (x + 4) (x + 4) = (x *x) + (x *4) + (4 *x) + (4 * 4) ;
= x² + 4x + 4x + 16 ;
= x² + 8x + 16;
→ Bring down the " − 46 " ;
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→ x² + 8x + 16 − 46 ;
= x² + 8x + 16 − 46 ;
= x² + 8x − 30 ;
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Now, rewrite the entire equation:
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→ x² + 2x = x² + 8x − 30 ;
Subtract "x² " & "2x" from EACH SIDE of the equation:
→ x² + 2x − x² − 2x = x² + 8x − 30 − x² − 2x ;
to get:
→ 0 = 6x − 30 ;
→ 6x − 30 = 0 ;
Add "30" to each side of the equation:
→ 6x − 30 + 30 = 0 + 30 ;
→ 6x = 30 ;
Divide each side of the equation by "6" ;
to isolate "x" on each side of the equation; & to solve for "x" ;
→ 6x / 6 = 30 / 6 ;
to get;
→ x = 5 ;
→ "(x + 2)" = "(5 + 2)" = 7 ;
→ "(x + 4)" = "(5 + 4)" = 9 ;
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Answer: The numbers are: " 5, 7, and 9 " .
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