Let's solve the given quadratic equation step by step to determine the true statement about the number of solutions.
The quadratic equation given is:
[tex]\[ y = x^2 - 11x + 7 \][/tex]
To determine the nature and number of the solutions, we need to use the discriminant, which is part of the quadratic formula. The discriminant [tex]\(\Delta\)[/tex] for a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Here, the coefficients are:
[tex]\[ a = 1, \, b = -11, \, c = 7 \][/tex]
Now, let's plug these values into the formula for the discriminant:
[tex]\[ \Delta = (-11)^2 - 4 \cdot 1 \cdot 7 \][/tex]
[tex]\[ \Delta = 121 - 28 \][/tex]
[tex]\[ \Delta = 93 \][/tex]
The discriminant is calculated to be 93.
The nature of the solutions depends on the value of the discriminant:
- If [tex]\(\Delta > 0\)[/tex] (positive), there are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex] (zero), there is exactly one real solution.
- If [tex]\(\Delta < 0\)[/tex] (negative), there are two complex (imaginary) solutions.
In our case, since [tex]\(\Delta = 93\)[/tex], which is greater than 0, this indicates that there are two distinct real solutions.
Therefore, the correct statement about the quadratic equation [tex]\( y = x^2 - 11x + 7 \)[/tex] is:
B. There are two real solutions.
