text: # Solve for t (4t^2)/5=(3t)/5+27/10
  
  4t25=3t5+2710
  Multiply both sides of the equation by 5.
  4t2=3t5⋅(5)+2710⋅(5)
  Simplify each term.
  Cancel the common factor of 5.
  Cancel the common factor.
  4t2=3t5⋅5+2710⋅(5)
  Rewrite the expression.
  4t2=3t+2710⋅(5)
  4t2=3t+2710⋅(5)
  Cancel the common factor of 5.
  Factor 5 out of 10.
  4t2=3t+275(2)⋅5
  Cancel the common factor.
  4t2=3t+275⋅2⋅5
  Rewrite the expression.
  4t2=3t+272
  4t2=3t+272
  4t2=3t+272
  Subtract 3t from both sides of the equation.
  4t2-3t=272
  Move 272 to the left side of the equation by subtracting it from both sides.
  4t2-3t-272=0
  Multiply through by the least common denominator 2, then simplify.
  Apply the distributive property.
  2(4t2)+2(-3t)+2(-272)=0
  Simplify.
  Multiply 4 by 2.
  8t2+2(-3t)+2(-272)=0
  Multiply -3 by 2.
  8t2-6t+2(-272)=0
  Cancel the common factor of 2.
  Move the leading negative in -272 into the numerator.
  8t2-6t+2(-272)=0
  Cancel the common factor.
  8t2-6t+2(-272)=0
  Rewrite the expression.
  8t2-6t-27=0
  8t2-6t-27=0
  8t2-6t-27=0
  8t2-6t-27=0
  Use the quadratic formula to find the solutions.
  -b±b2-4(ac)2a
  Substitute the values a=8, b=-6, and c=-27 into the quadratic formula and solve for t.
  6±(-6)2-4⋅(8⋅-27)2⋅8
  Simplify.
  Simplify the numerator.
  Raise -6 to the power of 2.
  t=6±36-4⋅(8⋅-27)2⋅8
  Multiply 8 by -27.
  t=6±36-4⋅-2162⋅8
  Multiply -4 by -216.
  t=6±36+8642⋅8
  t=6±9002⋅8
  Rewrite 900 as 302.
  t=6±3022⋅8
  Pull terms out from under the radical, assuming positive real numbers.
  t=6±302⋅8
  t=6±302⋅8
  Multiply 2 by 8.
  t=6±3016
  Simplify 6±3016.
  t=3±158
  t=3±158
  The final answer is the combination of both solutions.
  t=94,-32
  The result can be shown in multiple forms.
  Exact Form:
  t=94,-32
  Decimal Form:
  t=2.25,-1.5
  Mixed Number Form:
  t=214,-112
  Solve for t (4t^2)/5=(3t)/5+27/10
explanation: The document contains a detailed step-by-step solution to a quadratic equation. It starts with the equation \(\frac{4t^2}{5} = \frac{3t}{5} + \frac{27}{10}\) and proceeds to solve for \(t\). The solution includes simplifying the equation, moving terms, and applying the quadratic formula. The final answer is provided in exact, decimal, and mixed number forms.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Solve for t (4t^2)/5=(3t)/5+27/10
  
  4t25=3t5+2710
  Multiply both sides of the equation by 5.
  4t2=3t5⋅(5)+2710⋅(5)
  Simplify each term.
  Cancel the common factor of 5.
  Cancel the common factor.
  4t2=3t5⋅5+2710⋅(5)
  Rewrite the expression.
  4t2=3t+2710⋅(5)
  4t2=3t+2710⋅(5)
  Cancel the common factor of 5.
  Factor 5 out of 10.
  4t2=3t+275(2)⋅5
  Cancel the common factor.
  4t2=3t+275⋅2⋅5
  Rewrite the expression.
  4t2=3t+272
  4t2=3t+272
  4t2=3t+272
  Subtract 3t from both sides of the equation.
  4t2-3t=272
  Move 272 to the left side of the equation by subtracting it from both sides.
  4t2-3t-272=0
  Multiply through by the least common denominator 2, then simplify.
  Apply the distributive property.
  2(4t2)+2(-3t)+2(-272)=0
  Simplify.
  Multiply 4 by 2.
  8t2+2(-3t)+2(-272)=0
  Multiply -3 by 2.
  8t2-6t+2(-272)=0
  Cancel the common factor of 2.
  Move the leading negative in -272 into the numerator.
  8t2-6t+2(-272)=0
  Cancel the common factor.
  8t2-6t+2(-272)=0
  Rewrite the expression.
  8t2-6t-27=0
  8t2-6t-27=0
  8t2-6t-27=0
  8t2-6t-27=0
  Use the quadratic formula to find the solutions.
  -b±b2-4(ac)2a
  Substitute the values a=8, b=-6, and c=-27 into the quadratic formula and solve for t.
  6±(-6)2-4⋅(8⋅-27)2⋅8
  Simplify.
  Simplify the numerator.
  Raise -6 to the power of 2.
  t=6±36-4⋅(8⋅-27)2⋅8
  Multiply 8 by -27.
  t=6±36-4⋅-2162⋅8
  Multiply -4 by -216.
  t=6±36+8642⋅8
  t=6±9002⋅8
  Rewrite 900 as 302.
  t=6±3022⋅8
  Pull terms out from under the radical, assuming positive real numbers.
  t=6±302⋅8
  t=6±302⋅8
  Multiply 2 by 8.
  t=6±3016
  Simplify 6±3016.
  t=3±158
  t=3±158
  The final answer is the combination of both solutions.
  t=94,-32
  The result can be shown in multiple forms.
  Exact Form:
  t=94,-32
  Decimal Form:
  t=2.25,-1.5
  Mixed Number Form:
  t=214,-112
  Solve for t (4t^2)/5=(3t)/5+27/10
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a detailed step-by-step solution to a quadratic equation. It starts with the equation \(\frac{4t^2}{5} = \frac{3t}{5} + \frac{27}{10}\) and proceeds to solve for \(t\). The solution includes simplifying the equation, moving terms, and applying the quadratic formula. The final answer is provided in exact, decimal, and mixed number forms.
  
  Label: question-and-answer