Let's solve the quadratic equation step by step:
The given quadratic equation has the coefficients:
[tex]\[ a = 1, \, b = -6, \, c = 9 \][/tex]
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
1. Calculate the discriminant:
The discriminant (Δ) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values:
[tex]\[ \Delta = (-6)^2 - 4(1)(9) \][/tex]
[tex]\[ \Delta = 36 - 36 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
2. Calculate the roots using the quadratic formula:
Since the discriminant is 0, there is exactly one unique solution (repeated root):
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values:
[tex]\[ x = \frac{-(-6) \pm \sqrt{0}}{2(1)} \][/tex]
[tex]\[ x = \frac{6 \pm 0}{2} \][/tex]
[tex]\[ x = \frac{6}{2} \][/tex]
[tex]\[ x = 3 \][/tex]
Thus, the roots of the quadratic equation are:
[tex]\[ x = 3 \, \text{and} \, x = 3 \][/tex]
To summarize:
- The discriminant (Δ) is 0
- The solutions are [tex]\( x_1 = 3.0 \)[/tex] and [tex]\( x_2 = 3.0 \)[/tex]
These results confirm that the quadratic equation has one unique solution, repeated twice.
