Question

for questions 18 - 21, find the discriminant of the equation and determine the number and type of roots. solve each equation using the quadratic formula or factoring.

1. $3x^{2}+15x + 19 = 0$

discriminant:

/ type of roofs:

method (circle one):
f qf
solution(s):
© gina wilson (all things algebra®. llc), 2015 2022

Explanation:

Step1: Identify coefficients

For the quadratic equation \(ax^{2}+bx + c = 0\), in \(3x^{2}+15x + 19=0\), \(a = 3\), \(b = 15\), \(c = 19\).

Step2: Calculate discriminant

The discriminant formula is \(D=b^{2}-4ac\).
Substitute \(a = 3\), \(b = 15\), \(c = 19\) into the formula:

$$
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$$

Step3: Determine number of roots

Since \(D=-3<0\), the quadratic formula \(x=\frac{-b\pm\sqrt{D}}{2a}\) gives two complex roots.
Using the quadratic formula \(x=\frac{-15\pm\sqrt{-3}}{6}=\frac{-15\pm i\sqrt{3}}{6}\) (where \(i=\sqrt{-1}\))

Answer:

Discriminant: \(-3\)
\(\# /\) Type of Roots: \(2\) complex roots
Method: \(QF\) (Quadratic Formula)
Solution(s): \(x=\frac{-15 + i\sqrt{3}}{6}\) and \(x=\frac{-15 - i\sqrt{3}}{6}\)