To determine which line will have no solution with the parabola defined by the equation [tex]\( y - x + 2 = x^2 \)[/tex], we need to analyze the relationship between a general line equation and the parabola. Here is a detailed, step-by-step solution:
1. Equation of the Parabola:
The given parabola is described by the equation:
[tex]\[ y - x + 2 = x^2 \][/tex]
2. Equation of the Line:
A general line equation can be written as:
[tex]\[ y = mx + c \][/tex]
where [tex]\( m \)[/tex] is the slope and [tex]\( c \)[/tex] is the y-intercept.
3. Substitute the Line Equation into the Parabola's Equation:
Substitute [tex]\( y = mx + c \)[/tex] into [tex]\( y - x + 2 = x^2 \)[/tex]:
[tex]\[ (mx + c) - x + 2 = x^2 \][/tex]
Simplifying this, we get:
[tex]\[ mx + c - x + 2 = x^2 \][/tex]
4. Form a Quadratic Equation in Terms of [tex]\( x \)[/tex]:
Combine like terms to form a standard quadratic equation:
[tex]\[ x^2 + (m - 1)x + (c + 2) = 0 \][/tex]
This quadratic equation is in the form [tex]\( ax^2 + bx + c = 0 \)[/tex] with:
[tex]\[ a = 1, \quad b = m - 1, \quad c = c + 2 \][/tex]
5. Conditions for No Real Solutions:
For the line to not intersect the parabola, the quadratic equation must have no real solutions. This occurs when the discriminant of the quadratic equation is less than zero.
Remember, 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]
6. Calculate the Discriminant:
Substitute the coefficients [tex]\( a = 1 \)[/tex], [tex]\( b = m - 1 \)[/tex], and [tex]\( c = c + 2 \)[/tex] into the discriminant formula:
[tex]\[ \Delta = (m - 1)^2 - 4(1)(c + 2) \][/tex]
7. Discriminant Condition:
For the quadratic equation to have no real solutions, the discriminant must be less than zero:
[tex]\[ (m - 1)^2 - 4(c + 2) < 0 \][/tex]
8. Analyze the Discriminant Expression:
Since the discriminant is [tex]\((m - 1)^2\)[/tex], it’s apparent that [tex]\((m - 1)^2\)[/tex] is always non-negative (i.e., it’s either zero or positive for all real values of [tex]\( m \)[/tex] and [tex]\( c \)[/tex]). Therefore:
[tex]\[ (m - 1)^2 \ge 0 \][/tex]
Our condition for no real solutions, [tex]\((m - 1)^2 < 0\)[/tex], cannot be true as no real value squared gives a negative result. Thus, it leads us to the conclusion that such a line does not exist.
In summary, based on this analysis, there are no lines that have no solutions with the given parabola [tex]\( y - x + 2 = x^2 \)[/tex] since the discriminant condition cannot be satisfied for any real [tex]\( m \)[/tex] and [tex]\( c \)[/tex].
