Question

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

Quadratic function and equation
ANSWERED
asked 2020-12-06
For what values of m does the graph of \(\displaystyle{y}={3}{x}^{{2}}+{7}{x}+{m}\) have two x-intercepts

Answers (1)

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:
\(\displaystyle{m}{<}\frac{{49}}{{12}}\)

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