Answer: D. Graph D assuming that it is the last one shown.
Explanation
Given
[tex]y=3x²+7x+2[/tex]
we can determine the solutions of the equation (the points at which y = 0) and compare them with the graphs given to see which one is the correct one.
To solve the equation, we have to set it to 0:
[tex]0=3x²+7x+2[/tex]
Now, we can use the General Quadratic Formula to solve it:
[tex]x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}[/tex]
where a, b and c represent the coefficients of the equation in the form:
[tex]ax^2+bx+c=0[/tex]
Thus, in our case a = 3, b = 7, and c = 2. Replacing the values in the General Quadratic Formula and solving:
[tex]x_{1,2}=\frac{-7\pm\sqrt{7^2-4(3)(2)}}{2(3)}[/tex][tex]x_{1,2}=\frac{-7\pm\sqrt{49-24}}{6}[/tex][tex]x_{1,2}=\frac{-7\pm\sqrt{25}}{6}[/tex][tex]x_{1,2}=\frac{-7\pm5}{6}[/tex]
Finally, calculating our two solutions:
[tex]x_1=\frac{-7+5}{6}=\frac{-2}{6}=-\frac{1}{3}[/tex][tex]x_1=\frac{-7-5}{6}=\frac{-12}{6}=-2[/tex]
Based on these values, we can see that the graph that has two solutions in the negative numbers is:
