Question

question 15 of 23
use the discriminant to find possible solutions. do not solve.
$6x^2 - x = -14$
select the correct response:
2 real irrational solutions
2 non-real solutions
2 real rational solutions
1 real rational solution

Explanation:

Step1: Rewrite the equation in standard form

First, we need to rewrite the quadratic equation \(6x^{2}-x = - 14\) in the standard form \(ax^{2}+bx + c=0\). By adding 14 to both sides, we get \(6x^{2}-x + 14=0\). Here, \(a = 6\), \(b=-1\), and \(c = 14\).

Step2: Calculate the discriminant

The formula for the discriminant (\(D\)) of a quadratic equation \(ax^{2}+bx + c = 0\) is \(D=b^{2}-4ac\). Substitute \(a = 6\), \(b=-1\), and \(c = 14\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Analyze the discriminant

For a quadratic equation \(ax^{2}+bx + c = 0\):

• If \(D>0\), there are two distinct real solutions. If \(D\) is a perfect square, the solutions are rational; otherwise, they are irrational.
• If \(D = 0\), there is one real solution (a repeated root).
• If \(D<0\), there are two non - real (complex) solutions.

Since \(D=-335<0\), the quadratic equation has 2 non - real solutions.

Answer:

2 non - real solutions