Question

find the x-intercepts of the graph of the function $y = -3x^2 - 6x - 2$
$\bigcirc x = \dfrac{6 \pm \sqrt{60}}{-6}$
$\bigcirc x = \dfrac{6 \pm \sqrt{12}}{-3}$
$\bigcirc x = \dfrac{6 \pm \sqrt{12}}{-6}$
$\bigcirc x = \dfrac{-6 \pm \sqrt{12}}{-6}$
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Explanation:

Step1: Recall Quadratic Formula

For a quadratic equation \(ax^2 + bx + c = 0\), the solutions are \(x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\).
Given \(y=-3x^2 - 6x - 2\), to find x - intercepts, set \(y = 0\), so \(-3x^2-6x - 2=0\) or \(3x^2+6x + 2 = 0\) (multiplying both sides by - 1). Here, \(a = 3\), \(b = 6\), \(c = 2\). But we can also use the equation in the form \(-3x^2-6x - 2=0\) with \(a=-3\), \(b = - 6\), \(c=-2\).

Step2: Apply Quadratic Formula

Using \(a=-3\), \(b=-6\), \(c = - 2\) in \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\):
First, calculate \(b^2-4ac\):
\(b^2-4ac=(-6)^2-4\times(-3)\times(-2)=36 - 24 = 12\)
Then, \(-b=6\), \(2a=2\times(-3)=-6\)
So \(x=\frac{6\pm\sqrt{12}}{-6}\)

Answer:

\(x=\frac{6\pm\sqrt{12}}{-6}\) (the third option)