To determine the number of solutions for the quadratic equation [tex]\(x^2 = -7x + 7\)[/tex], we need to follow these steps:
1. Rewrite the Equation: Rewrite the given quadratic equation in standard form [tex]\(ax^2 + bx + c = 0\)[/tex].
[tex]\[ x^2 + 7x - 7 = 0 \][/tex]
2. Identify Coefficients: Identify the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] from the equation [tex]\(ax^2 + bx + c = 0\)[/tex].
[tex]\[ a = 1, \, b = -7, \, c = 7 \][/tex]
3. Calculate the Discriminant: Use the formula for the discriminant [tex]\(\Delta = b^2 - 4ac\)[/tex].
[tex]\[ \Delta = (-7)^2 - 4(1)(7) \][/tex]
[tex]\[ \Delta = 49 - 28 \][/tex]
[tex]\[ \Delta = 21 \][/tex]
4. Determine the Number of Solutions: Based on the value of the discriminant [tex]\(\Delta\)[/tex]:
- 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 no real solutions.
Since the discriminant [tex]\(\Delta = 21\)[/tex] is greater than 0, we conclude that the quadratic equation has two distinct real solutions.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{c. two solutions}} \][/tex]
