The value of the discriminant is 25. There are two x-intercepts and two real zeros.
How to derive the characteristics of a second order polynomial by quadratic formula
All roots of second order polynomial of the form \(a\cdot x^{2}+b\cdot x + c = 0\) can be be found analytically by quadratic formula:
\(x_{1,2} = \frac{-b\pm \sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}\), \(a, b, c\in \mathbb{R}\) (1)
Where \(b^{2}-4\cdot a\cdot c\) is the discriminant. There are three scenarios:
- If discriminant is greater than zero, then the polynomial has two different real x-intercepts. Two real zeros.
- If discriminant is equal to zero, then the polynomial has two equal real x-intercepts. Two real zeros.
- If discriminant is less that zero, then the polynomial has no x-intercepts. Two conjugated complex zeros.
If we know that \(a = 3\), \(b = 7\) and \(c = 2\), then the discriminant is:
\(d = 7^{2}-4\cdot (3)\cdot (2)\)
\(d = 49-24\)
\(d = 25\)
The value of the discriminant is 25. There are two x-intercepts and two real zeros. \(\blacksquare\)
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