Use partial factoring to determine the vertex of y=3x^2+5x-1.

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To determine the vertex of the quadratic function y = 3x^2 + 5x - 1 using partial factoring, we can follow the steps below:

Step 1: Write the quadratic function in the standard form: y = ax^2 + bx + c, where "a", "b", and "c" are the coefficients of the quadratic equation.

In this case, the equation is already in the standard form: y = 3x^2 + 5x - 1.

Step 2: Identify the values of "a" and "b" from the quadratic function.

From the given equation, we can see that "a" is the coefficient of x^2 term, which is 3, and "b" is the coefficient of the x term, which is 5.

Step 3: Find the x-coordinate of the vertex using the formula: x = -b / (2a).

Substituting the values, we have: x = -5 / (2 * 3) = -5/6.

Step 4: Calculate the corresponding y-coordinate of the vertex by substituting the x-coordinate into the original equation.

Substituting x = -5/6 into the equation y = 3x^2 + 5x - 1, we have:

y = 3*(-5/6)^2 + 5*(-5/6) - 1
= 3*(25/36) - 25/6 - 1
= 75/36 - 25/6 - 1
= (75 - 150 - 36) / 36
= -111/36
= -37/12

Therefore, the vertex of the quadratic function y = 3x^2 + 5x - 1 is (-5/6, -37/12).