To determine which line will have no solutions when intersecting the parabola given by the equation [tex]\( y - x + 2 = x^2 \)[/tex], we need to find a line such that the system of equations has no intersections. This involves considering the quadratic equation formed when substituting the line equation into the parabola equation.
### Step-by-Step Solution:
1. Rewrite the Parabola Equation:
[tex]\[
y - x + 2 = x^2 \implies y = x^2 + x - 2
\][/tex]
2. Consider the Line Equation:
[tex]\[
y = mx + c
\][/tex]
3. Substitute the Line Equation into the Parabola Equation:
Substitute [tex]\( y = mx + c \)[/tex] into [tex]\( y = x^2 + x - 2 \)[/tex]:
[tex]\[
mx + c = x^2 + x - 2
\][/tex]
4. Form the Quadratic Equation:
Rearrange the equation to form a standard quadratic equation:
[tex]\[
x^2 + x - mx - 2 - c = 0 \implies x^2 + (1 - m)x - (2 + c) = 0
\][/tex]
This quadratic equation is of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
[tex]\[
a = 1, \quad b = 1 - m, \quad c = -(2 + c)
\][/tex]
5. Compute 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 [tex]\( a = 1 \)[/tex], [tex]\( b = 1 - m \)[/tex], and [tex]\( c = -(2 + c) \)[/tex] into the discriminant formula:
[tex]\[
\Delta = (1 - m)^2 - 4 \cdot 1 \cdot -(2 + c) = (1 - m)^2 + 4(2 + c)
\][/tex]
6. Set the Discriminant to Be Less than Zero:
For the quadratic equation to have no real solutions, the discriminant must be less than zero:
[tex]\[
(1 - m)^2 + 4(2 + c) < 0
\][/tex]
7. Simplify the Inequality:
[tex]\[
(1 - m)^2 + 4(2 + c) < 0 \implies 1 - 2m + m^2 + 8 + 4c < 0 \implies m^2 - 2m + 9 + 4c < 0
\][/tex]
8. Solve for the Condition on [tex]\( c \)[/tex]:
[tex]\[
4c < - (m^2 - 2m + 9) \implies 4c < - m^2 + 2m - 9 \implies c < \frac{- m^2 + 2m - 9}{4}
\][/tex]
Thus, the line [tex]\( y = mx + c \)[/tex] will have no intersection with the parabola [tex]\( y - x + 2 = x^2 \)[/tex] if and only if the constant [tex]\( c \)[/tex] satisfies the inequality:
[tex]\[
c < \frac{- m^2 + 2m - 9}{4}
\][/tex]
