To determine the number of solutions to the equation [tex]\( 6x - 2x^2 = 7 \)[/tex], we first rewrite it in standard form. The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex]. We proceed as follows:
1. Start with the given equation:
[tex]\[
6x - 2x^2 = 7
\][/tex]
2. Move all terms to one side of the equation to set it to zero:
[tex]\[
-2x^2 + 6x - 7 = 0
\][/tex]
Next, let's identify the coefficients in the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
- [tex]\( a = -2 \)[/tex]
- [tex]\( b = 6 \)[/tex]
- [tex]\( c = -7 \)[/tex]
Now, we calculate 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]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula:
[tex]\[
\Delta = (6)^2 - 4(-2)(-7)
\][/tex]
[tex]\[
\Delta = 36 - 56
\][/tex]
[tex]\[
\Delta = -20
\][/tex]
The number of solutions of the quadratic equation 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 no real solutions.
Since the discriminant [tex]\(\Delta = -20\)[/tex] is less than zero, there are no real solutions to the given quadratic equation.
Thus, the given polynomial has exactly [tex]\(\boxed{0}\)[/tex] solution(s).
