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 [tex]a\cdot x^{2}+b\cdot x + c = 0[/tex] can be be found analytically by quadratic formula:
[tex]x_{1,2} = \frac{-b\pm \sqrt{b^{2}-4\cdot a\cdot c}}{2\cdot a}[/tex], [tex]a, b, c\in \mathbb{R}[/tex] (1)
Where [tex]b^{2}-4\cdot a\cdot c[/tex] 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 [tex]a = 3[/tex], [tex]b = 7[/tex] and [tex]c = 2[/tex], then the discriminant is:
[tex]d = 7^{2}-4\cdot (3)\cdot (2)[/tex]
[tex]d = 49-24[/tex]
[tex]d = 25[/tex]
The value of the discriminant is 25. There are two x-intercepts and two real zeros. [tex]\blacksquare[/tex]
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