Question

how many x-intercepts does the parabola with the following equation have?
y = 6x² + 18x + 6
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Explanation:

Step1: Recall the discriminant formula for a quadratic equation \(ax^2 + bx + c = 0\) (where \(y = 0\) for x - intercepts). The discriminant \(D=b^{2}-4ac\). For the parabola \(y = 6x^{2}+18x + 6\), when finding x - intercepts, we set \(y = 0\), so the quadratic equation is \(6x^{2}+18x + 6=0\). Here, \(a = 6\), \(b = 18\), \(c = 6\).

Step2: Calculate the discriminant. Substitute the values of \(a\), \(b\), and \(c\) into the discriminant formula: \(D=(18)^{2}-4\times6\times6\). First, calculate \((18)^{2}=324\) and \(4\times6\times6 = 144\). Then, \(D = 324-144=180\).

Step3: Analyze the discriminant. Since the discriminant \(D=180>0\), a quadratic equation \(ax^{2}+bx + c = 0\) with \(D>0\) has two distinct real roots. These roots correspond to the x - intercepts of the parabola.

Answer:

2