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Answer:
Step-by-step explanation:
The quadratic expression can be factored into two binomials if it has two real roots.
Two real roots possible with non-negative discriminant:
As D = b² - 4ac, we get the following inequality
This is true for any value of k
Answer:
(-∞, ∞) or [tex]k \in \mathbb{R}[/tex]
Step-by-step explanation:
Binomial: two terms connected by a plus or minus sign.
Discriminant
[tex]b^2-4ac\quad\textsf{when}\:ax^2+bx+c=0[/tex]
[tex]\textsf{when }\:b^2-4ac > 0 \implies \textsf{two real roots}[/tex]
[tex]\textsf{when }\:b^2-4ac=0 \implies \textsf{one real root}[/tex]
[tex]\textsf{when }\:b^2-4ac < 0 \implies \textsf{no real roots}[/tex]
If a quadratic expression factors into two binomials, it will have two real roots. Therefore, the discriminant will be greater than zero.
Given quadratic expression:
[tex]3x^2+kx-8[/tex]
[tex]\implies a=3, \quad b=k, \quad c=-8[/tex]
Substitute the values of a, b and c into the discriminant, set it to > 0:
[tex]\implies k^2-4(3)(-8) > 0[/tex]
[tex]\implies k^2+96 > 0[/tex]
As k² ≥ 0 for all real numbers,
[tex]\implies k^2+96 \geq 96[/tex]
Therefore, the values of k are (-∞, ∞) or [tex]k \in \mathbb{R}[/tex]