# 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
The text passage contains a detailed step-by-step solution to a quadratic equation, including the problem statement and the final answer in multiple forms.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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
user
The text passage contains a detailed step-by-step solution to a quadratic equation, including the problem statement and the final answer in multiple forms.

TYPE: SINGLE_QUESTION_WITH_ANSWER