1) 2 - 18a^2 - a = -3a^2 Rewriting the equation: -18a^2 - a + 3a^2 + 2 = 0 Combining like terms: -15a^2 - a + 2 = 0 We can solve this quadratic equation by factoring: (-3a+2)(5a-1) = 0 Setting each factor to zero: -3a + 2 = 0 or 5a - 1 = 0 -3a = -2 5a = 1 a = 2/3 a = 1/5
Therefore, the solutions to the first equation are a = 2/3 and a = 1/5.
2) 5a + 7 - 28a^2 = a^2 Rearranging the equation: 5a + 7 = 29a^2 Subtracting 5a and 7 from both sides: 29a^2 - 5a - 7 = 0 Since this is a quadratic equation, we can use the quadratic formula to solve for 'a': a = [-(-5) ± sqrt((-5)^2 - 429(-7))]/(2*29) a = (5 ± sqrt(25 + 812))/58 a = (5 ± √837)/58
Therefore, the solutions to the second equation are a = (5 + √837)/58 and a = (5 - √837)/58.
Let's solve each equation separately:
1) 2 - 18a^2 - a = -3a^2
Rewriting the equation:
-18a^2 - a + 3a^2 + 2 = 0
Combining like terms:
-15a^2 - a + 2 = 0
We can solve this quadratic equation by factoring:
(-3a+2)(5a-1) = 0
Setting each factor to zero:
-3a + 2 = 0 or 5a - 1 = 0
-3a = -2 5a = 1
a = 2/3 a = 1/5
Therefore, the solutions to the first equation are a = 2/3 and a = 1/5.
2) 5a + 7 - 28a^2 = a^2
Rearranging the equation:
5a + 7 = 29a^2
Subtracting 5a and 7 from both sides:
29a^2 - 5a - 7 = 0
Since this is a quadratic equation, we can use the quadratic formula to solve for 'a':
a = [-(-5) ± sqrt((-5)^2 - 429(-7))]/(2*29)
a = (5 ± sqrt(25 + 812))/58
a = (5 ± √837)/58
Therefore, the solutions to the second equation are a = (5 + √837)/58 and a = (5 - √837)/58.