Answer :
To find the value of the slope [tex]\( m \)[/tex] such that the line [tex]\( y = mx - 3 \)[/tex] is tangent to the parabola [tex]\( y = 3x^2 - x \)[/tex], we need to determine the condition under which the line only intersects the parabola at a single point. This typically involves solving the system of equations given by setting the two expressions for [tex]\( y \)[/tex] equal to each other and ensuring that the resulting quadratic equation has exactly one solution (i.e., its discriminant is zero).
### Step-by-Step Solution
1. Set the equations equal to each other:
[tex]\[ mx - 3 = 3x^2 - x \][/tex]
2. Rearrange the equation to standard quadratic form:
[tex]\[ 3x^2 - (m + 1)x + 3 = 0 \][/tex]
3. Identify the coefficients of the quadratic equation:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Comparing with the standard form, we have:
[tex]\[ a = 3, \quad b = -(m + 1), \quad c = 3 \][/tex]
4. Recall the formula for the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
5. Substitute the coefficients into the discriminant formula:
[tex]\[ \Delta = [-(m + 1)]^2 - 4 \cdot 3 \cdot 3 \][/tex]
6. Simplify the discriminant:
[tex]\[ \Delta = (m + 1)^2 - 36 \][/tex]
7. For the line to be tangent to the parabola, the discriminant must be zero:
[tex]\[ (m + 1)^2 - 36 = 0 \][/tex]
8. Solve for [tex]\( m \)[/tex]:
[tex]\[ (m + 1)^2 = 36 \][/tex]
9. Take the square root of both sides:
[tex]\[ m + 1 = \pm 6 \][/tex]
10. Solve the resulting equations:
[tex]\[ m + 1 = 6 \quad \text{or} \quad m + 1 = -6 \][/tex]
[tex]\[ m = 5 \quad \text{or} \quad m = -7 \][/tex]
### Conclusion
The slopes [tex]\( m \)[/tex] for which the line [tex]\( y = mx - 3 \)[/tex] is tangent to the parabola [tex]\( y = 3x^2 - x \)[/tex] are [tex]\( \boxed{5 \text{ and } -7} \)[/tex].
### Step-by-Step Solution
1. Set the equations equal to each other:
[tex]\[ mx - 3 = 3x^2 - x \][/tex]
2. Rearrange the equation to standard quadratic form:
[tex]\[ 3x^2 - (m + 1)x + 3 = 0 \][/tex]
3. Identify the coefficients of the quadratic equation:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Comparing with the standard form, we have:
[tex]\[ a = 3, \quad b = -(m + 1), \quad c = 3 \][/tex]
4. Recall the formula for the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
5. Substitute the coefficients into the discriminant formula:
[tex]\[ \Delta = [-(m + 1)]^2 - 4 \cdot 3 \cdot 3 \][/tex]
6. Simplify the discriminant:
[tex]\[ \Delta = (m + 1)^2 - 36 \][/tex]
7. For the line to be tangent to the parabola, the discriminant must be zero:
[tex]\[ (m + 1)^2 - 36 = 0 \][/tex]
8. Solve for [tex]\( m \)[/tex]:
[tex]\[ (m + 1)^2 = 36 \][/tex]
9. Take the square root of both sides:
[tex]\[ m + 1 = \pm 6 \][/tex]
10. Solve the resulting equations:
[tex]\[ m + 1 = 6 \quad \text{or} \quad m + 1 = -6 \][/tex]
[tex]\[ m = 5 \quad \text{or} \quad m = -7 \][/tex]
### Conclusion
The slopes [tex]\( m \)[/tex] for which the line [tex]\( y = mx - 3 \)[/tex] is tangent to the parabola [tex]\( y = 3x^2 - x \)[/tex] are [tex]\( \boxed{5 \text{ and } -7} \)[/tex].
