To determine the nature of the solutions for the quadratic equation [tex]\( y = x^2 - 11x + 7 \)[/tex], you need to examine its discriminant.
The general form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. For the given equation [tex]\( x^2 - 11x + 7 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -11 \)[/tex]
- [tex]\( c = 7 \)[/tex]
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation is calculated using the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Substituting the given coefficients into the formula:
[tex]\[
\Delta = (-11)^2 - 4(1)(7)
\][/tex]
Calculate [tex]\((-11)^2\)[/tex]:
[tex]\[
(-11)^2 = 121
\][/tex]
Now multiply [tex]\( 4 \times 1 \times 7 \)[/tex]:
[tex]\[
4 \times 1 \times 7 = 28
\][/tex]
Next, subtract these values:
[tex]\[
121 - 28 = 93
\][/tex]
So, the discriminant for this quadratic equation is [tex]\( \Delta = 93 \)[/tex].
The nature of the solutions depends on the value of the discriminant:
- If [tex]\(\Delta > 0\)[/tex], there are two distinct real solutions.
- If [tex]\(\Delta = 0\)[/tex], there is exactly one real solution.
- If [tex]\(\Delta < 0\)[/tex], there are two complex solutions.
Since the discriminant [tex]\(\Delta = 93\)[/tex] is greater than zero, the quadratic equation [tex]\( x^2 - 11x + 7 = 0 \)[/tex] has two distinct real solutions.
Therefore, the correct answer is:
C. There are two real solutions.
