To determine for which values of [tex]\(k\)[/tex] the quadratic equation [tex]\(2x^2 - 6x + k = 0\)[/tex] has no real roots, we need to analyze the discriminant ([tex]\(\Delta\)[/tex]) of the quadratic equation.
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. For our equation:
- [tex]\(a = 2\)[/tex]
- [tex]\(b = -6\)[/tex]
- [tex]\(c = k\)[/tex]
The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
For the quadratic equation to have no real roots, the discriminant must be less than 0:
[tex]\[
\Delta < 0
\][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula, we get:
[tex]\[
\Delta = (-6)^2 - 4(2)(k)
\][/tex]
Simplify the expression inside the discriminant:
[tex]\[
\Delta = 36 - 8k
\][/tex]
We set up the inequality for no real roots:
[tex]\[
36 - 8k < 0
\][/tex]
Solving this inequality for [tex]\(k\)[/tex]:
First, subtract 36 from both sides:
[tex]\[
-8k < -36
\][/tex]
Next, divide both sides by -8 (remember that dividing by a negative number reverses the inequality sign):
[tex]\[
k > \frac{36}{8}
\][/tex]
Simplify the fraction:
[tex]\[
k > 4.5
\][/tex]
Therefore, the quadratic equation [tex]\(2x^2 - 6x + k = 0\)[/tex] has no real roots when [tex]\(k > 4.5\)[/tex].
