Answer :
To determine the number of solutions for the quadratic equation [tex]\(2x^2 + 8x + 8 = 0\)[/tex], we need to compute and analyze the discriminant. The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
where [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] are the coefficients of the equation. In this case:
[tex]\[ a = 2, \quad b = 8, \quad c = 8 \][/tex]
Plugging these values into the formula for the discriminant, we get:
[tex]\[ \Delta = 8^2 - 4 \cdot 2 \cdot 8 \][/tex]
Calculating the discriminant step-by-step:
[tex]\[ 8^2 = 64 \][/tex]
and
[tex]\[ 4 \cdot 2 \cdot 8 = 64 \][/tex]
So,
[tex]\[ \Delta = 64 - 64 = 0 \][/tex]
The discriminant [tex]\(\Delta\)[/tex] is equal to 0. The value of the discriminant determines the number and nature of the solutions:
- If [tex]\(\Delta > 0\)[/tex], the equation has 2 distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly 1 real solution.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real solutions.
Since the discriminant [tex]\(\Delta = 0\)[/tex] in this scenario, the quadratic equation [tex]\(2x^2 + 8x + 8 = 0\)[/tex] has exactly 1 real solution.
Thus, the correct answer is:
[tex]\[ \boxed{1 \text{ real solution}} \][/tex]
[tex]\[ \Delta = b^2 - 4ac \][/tex]
where [tex]\( a, b, \)[/tex] and [tex]\( c \)[/tex] are the coefficients of the equation. In this case:
[tex]\[ a = 2, \quad b = 8, \quad c = 8 \][/tex]
Plugging these values into the formula for the discriminant, we get:
[tex]\[ \Delta = 8^2 - 4 \cdot 2 \cdot 8 \][/tex]
Calculating the discriminant step-by-step:
[tex]\[ 8^2 = 64 \][/tex]
and
[tex]\[ 4 \cdot 2 \cdot 8 = 64 \][/tex]
So,
[tex]\[ \Delta = 64 - 64 = 0 \][/tex]
The discriminant [tex]\(\Delta\)[/tex] is equal to 0. The value of the discriminant determines the number and nature of the solutions:
- If [tex]\(\Delta > 0\)[/tex], the equation has 2 distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], the equation has exactly 1 real solution.
- If [tex]\(\Delta < 0\)[/tex], the equation has no real solutions.
Since the discriminant [tex]\(\Delta = 0\)[/tex] in this scenario, the quadratic equation [tex]\(2x^2 + 8x + 8 = 0\)[/tex] has exactly 1 real solution.
Thus, the correct answer is:
[tex]\[ \boxed{1 \text{ real solution}} \][/tex]
