Question

# For what values of m does the graph of y=3x^2+7x+m have two x-intercepts

For what values of m does the graph of $$\displaystyle{y}={3}{x}^{{2}}+{7}{x}+{m}$$ have two x-intercepts

2020-12-07

Given
$$\displaystyle{y}={3}{x}^{{2}}+{7}{x}+{m}$$
The graph of a quadratic function has two x-intecepts when the discriminant of the equation is positive.
The discriminant of a quadratic equation $$\displaystyle{y}={a}{x}^{{2}}+{b}{x}+{c}\ {i}{s}\ {D}={b}^{{2}}-{4}{a}{c}.$$
$$D=b^2-4ac =7^2-4(3)(m) =49-12m$$
The discriminant needs to be positive when the graph has two x-intecepts:
49-12m>0
Subtract 49 from each side of the inequality:
-12m>-49
Divide each side of the inequality by -12
$$\displaystyle{m}{<}{\left(-\frac{{49}}{{-{{12}}}}\right)}$$
Simplify:
$$\displaystyle{m}{<}\frac{{49}}{{12}}$$