Answer :
To determine the number of solutions for the quadratic equation [tex]\(x^2 - 4x + 4 = 0\)[/tex], we need to analyze its discriminant. The discriminant ([tex]\(\Delta\)[/tex]) for a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In our specific equation [tex]\(x^2 - 4x + 4 = 0\)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 4\)[/tex]
Substituting these values into the discriminant formula, we get:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 4 \][/tex]
Calculating this step-by-step:
[tex]\[ \Delta = 16 - 16 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
The discriminant ([tex]\(\Delta\)[/tex]) is 0. The number of solutions of a quadratic equation depends on the value of the discriminant:
1. If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real solutions.
2. If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real solution (often called a repeated or double root).
3. If [tex]\(\Delta < 0\)[/tex], the equation has no real solutions but two complex solutions.
Since our discriminant is 0, the quadratic equation [tex]\(x^2 - 4x + 4 = 0\)[/tex] has exactly one real solution.
Thus, the correct answer is:
(B) 1 real solution
[tex]\[ \Delta = b^2 - 4ac \][/tex]
In our specific equation [tex]\(x^2 - 4x + 4 = 0\)[/tex], the coefficients are:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = -4\)[/tex]
- [tex]\(c = 4\)[/tex]
Substituting these values into the discriminant formula, we get:
[tex]\[ \Delta = (-4)^2 - 4 \cdot 1 \cdot 4 \][/tex]
Calculating this step-by-step:
[tex]\[ \Delta = 16 - 16 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
The discriminant ([tex]\(\Delta\)[/tex]) is 0. The number of solutions of a quadratic equation depends on the value of the discriminant:
1. If [tex]\(\Delta > 0\)[/tex], the equation has two distinct real solutions.
2. If [tex]\(\Delta = 0\)[/tex], the equation has exactly one real solution (often called a repeated or double root).
3. If [tex]\(\Delta < 0\)[/tex], the equation has no real solutions but two complex solutions.
Since our discriminant is 0, the quadratic equation [tex]\(x^2 - 4x + 4 = 0\)[/tex] has exactly one real solution.
Thus, the correct answer is:
(B) 1 real solution
