Answer:
we can find a + a^(1/2) by substitution:
a + a^(1/2) = 6/5 + √(6/5)
Explanation:
We have the equation:
a² + a - 1/2 = 5
Adding 1/2 to both sides:
a² + a = 5 1/2
Factoring out a from the left-hand side:
a(a + 1) = 5 1/2
Taking the square root of both sides:
a + 1/2 = ±√(5 1/2 / a)
Since a^(1/2) - a^(1/2) > 0, we know that a is positive, so we can discard the negative root. Then we have:
a + 1/2 = √(5 1/2 / a)
Squaring both sides:
a² + a + 1/4 = 5 1/2 / a
Substituting the expression for a² + a from the first equation:
5 1/2 - 1/2 + 1/4 = 5 1/2 / a
Multiplying both sides by a:
5 1/2 a - 1/2 a + 1/4 a = 5 1/2
Simplifying:
5a - 1/2 = 5 1/2
Adding 1/2 to both sides:
5a = 6
Dividing by 5:
a = 6/5
Finally, we can find a + a^(1/2) by substitution:
a + a^(1/2) = 6/5 + √(6/5)