olve by factoring.5tsquaredminus36tequals32Question content area bottomPart 1Select the correct choice and fill in any answer boxes in your choice below.A.tequalsenter your response here(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as needed.)B.The solution is not a real number.

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olve by factoring.5tsquaredminus36tequals32Question content area bottomPart 1Select the correct choice and fill in any answer boxes in your choice below.A.tequalsenter your response here​(Simplify your answer. Type an integer or a fraction. Use a comma to separate answers as​ needed.)B.The solution is not a real number.

GPT-4o mini · Answer

To solve the equation $ 5t^2 - 36t = 32 $ by factoring, we first rearrange it to set it to zero: $$ 5t^2 - 36t - 32 = 0 $$ Next, we can try to factor the quadratic equation $ 5t^2 - 36t - 32 $. We are looking for two numbers that multiply to $ 5 \times -32 = -160 $ and add up to $-36$. The pairs of factors of $-160$ that could work would need to be checked to see which sum is $-36$. After checking various pairs, we find that: $$ -40 \text{ and } 4 $$ multiply to $-160$ and add to $-36$. We can now rewrite the middle term using these two factors: $$ 5t^2 - 40t + 4t - 32 = 0 $$ Next, we group the terms: $$ (5t^2 - 40t) + (4t - 32) = 0 $$ Factoring each group gives: $$ 5t(t - 8) + 4(t - 8) = 0 $$ Now we can factor by grouping: $$ (t - 8)(5t + 4) = 0 $$ Setting each factor to zero gives us: 1. $ t - 8 = 0 $ → $ t = 8 $ 2. $ 5t + 4 = 0 $ → $ 5t = -4 $ → $ t = -\frac{4}{5} $ So the solutions are: $$ t = 8, -\frac{4}{5} $$ Finally, the answer in simplified form is: **A. $ t = 8, -\frac{4}{5} $**.