Determine the zero(s) of the function `y=-3x^{2}+4x+11`

Question

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Answer 1

To find the zeros of the function, we need to solve the equation y = -3x^2 + 4x + 11 for x when y is equal to zero.

So, we have -3x^2 + 4x + 11 = 0.

To solve this quadratic equation, we can try factoring or using the quadratic formula.
Since factoring is not apparent, we can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a.

For our equation -3x^2 + 4x + 11 = 0, a = -3, b = 4, and c = 11.

Using the quadratic formula, we have:

x = (-4 ± sqrt(4^2 - 4(-3)(11))) / 2(-3)
= (-4 ± sqrt(16 + 132)) / (-6)
= (-4 ± sqrt(148)) / (-6)
= (-4 ± sqrt(4 * 37)) / (-6)
= (-4 ± 2sqrt(37)) / (-6).

So, the zeros of the function are:
x = (-4 + 2sqrt(37)) / (-6) and x = (-4 - 2sqrt(37)) / (-6).

These are the two solutions to the quadratic equation, which represent the x-intercepts or zeros of the function.