Final answer:
The equivalent equation to the given equation 12b(2b-3)=7 is obtained by simplifying and reformatting the original equation. The correct form that Rick can solve with the same solution as the original equation is (b - 34)² = 103/16, which corresponds to option 4.
Step-by-step explanation:
When looking for an equation that has the same solution as the given equation 12b(2b-3)=7, we're searching for an equation that is mathematically equivalent. This involves simplifying the original equation and finding a form that's easier to solve. To begin with, we should expand the left-hand side and divide both sides of the equation by 12, which will give us a quadratic equation in terms of b. Let's go step by step:
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- First expand: 12b(2b) - 12b(3) = 7, which simplifies to 24b² - 36b = 7.
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- Then, divide everything by 12: 24b²/12 - 36b/12 = 7/12, which simplifies to 2b² - 3b = 7/12.
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- To make it look like the equations in the options we're given, we need to get everything over a common denominator and set it to zero: 2b² - 3b - 7/12 = 0.
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- Now this equation can be multiplied through by 12 to clear the fraction: 24b² - 36b - 7 = 0.
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- Finally, we look for expressions that factor into this form among the options provided. Option 4) (b - 34)² = 103/16 when expanded and multiplied by 16, gives us this form.
Therefore, the equivalent equation Rick can solve that will have the same solution as the original equation is option 4) (b - 34)² = 103/16.