Solve 0 = 3x^2 + 5x -1 by completing the square. Express your answer as exact roots.

3(x^2 + 5/3 + 25/36) = -1
(x+ 5/6)^2 = 37/36
sq root( x + 5/6)^2 = sq root (37/36)
x + 5/6 = +- sq root 37/6

x = -5/6 +- square root 37 /6 <---

(Sorry for the work being so messy! It'd be much neater if I could type the symbol for square root)

You are correct,

but write it as
x = (-5 ± √37)/6

I use a Mac and to get √7 I press "option v"

Thanks! But could you please explain why we take off one of the 6's?

I use a Windows PC and cant find the command for square roots

Because she used parentheses.

Thank you

No problem! Your steps are correct. To solve the quadratic equation 0 = 3x^2 + 5x - 1 by completing the square, you followed these steps:

1. Begin with the given quadratic equation: 3x^2 + 5x - 1 = 0.
2. Divide the equation by 3 so that the coefficient of x^2 is 1: x^2 + (5/3)x - 1/3 = 0.
3. Move the constant term (-1/3) to the right side of the equation to isolate the x terms: x^2 + (5/3)x = 1/3.
4. To complete the square, add the square of half of the coefficient of x to both sides. In this case, half of (5/3) is 5/6, so we add (5/6)^2 to both sides: x^2 + (5/3)x + (5/6)^2 = 1/3 + (5/6)^2.
5. Simplify the right side of the equation: x^2 + (5/3)x + (25/36) = 1/3 + 25/36.
6. Combine the right side fractions by finding a common denominator: x^2 + (5/3)x + (25/36) = (12/36) + (25/36).
7. Add the fractions on the right side: x^2 + (5/3)x + (25/36) = 37/36.
8. Factor the left side as a perfect square trinomial: (x + 5/6)^2 = 37/36.
9. Take the square root of both sides: square root((x + 5/6)^2) = square root(37/36).
10. Simplify the square root on the right side: x + 5/6 = ±√(37/36).
11. Subtract 5/6 from both sides to isolate x: x = -5/6 ± √(37/36).

So the solution to the quadratic equation is x = -5/6 ± √(37)/6.