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jacob193

Answer:

D) [tex]x^{2} + 2x + 3[/tex].

Step-by-step explanation:

All four polynomials are quadratic, meaning that the highest power of the unknown [tex]x[/tex] in the equation is two.

The sign of the quadratic discriminant, [tex]\Delta[/tex], is a way to tell if a quadratic polynomial comes with non-real solutions.

There are three cases:

  • [tex]\Delta > 0[/tex]. The quadratic discriminant is positive. There are two real solutions and no non-real solution. The two solutions are different from each other.
  • [tex]\Delta = 0[/tex]. The quadratic discriminant is zero. There is one real solution and no non-real solution.
  • [tex]\Delta <0[/tex]. The quadratic discriminant is negative. There is no real solution and two non-real solutions.

How to find the quadratic discriminant?

If the equation is in this form:

[tex]a \; x^{2} + b\;x + c = 0[/tex],

where a, b, and c are real numbers (a.k.a. "constants.")

Quadratic discriminant:

[tex]\Delta = {b^{2} - 4\;a\cdot c}[/tex].

Polynomial in A:

[tex]x^{2} - 6x + 3 = 0[/tex].

  • [tex]a = 1[/tex].
  • [tex]b = -6[/tex].
  • [tex]c = 3[/tex].

[tex]\Delta = b^{2} - 4 \;a\cdot c = (-6)^{2} - 4\times 1\times 3 = 36 - 12 = 24[/tex].

[tex]\Delta > 0[/tex]. There will be no non-real solutions and two distinct real solutions.

Try the steps above for the polynomial in B, C, and D.

  • B): [tex]\Delta = 4[/tex]. Two distinct real solutions. No non-real solution.
  • C): [tex]\Delta= 41[/tex]. Two distinct real solutions. No non-real solution.
  • D): [tex]\Delta = -8[/tex]. No real solution. Two distinct non-real solutions.

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