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Find all the values of k so that the quadratic expression factors into two binomials. Explain the process used to find the values.

3x^2 + kx - 8

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mhanifa

Answer:

  • Any value of k

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:

  • D ≥ 0

As D = b² - 4ac, we get the following inequality

  • k² - 4(3)(-8) ≥ 0
  • k² + 96 ≥ 0
  • k² ≥ - 96

This is true for any value of k

  • k ∈ (- ∞, + ∞)
semsee45

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]

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