Determine the value(s) of m so that the line y = mx - 30 never intersects the parabola y = 2x² + 12x + 2. [-4 < m < 28]
Understand the Problem
The question is asking to determine the values of m such that the line represented by the equation y = mx - 30 does not intersect the parabola given by y = 2x² + 12x + 2. This involves analyzing the conditions under which the quadratic equation formed by setting the two equations equal has no real solutions.
Answer
The values of \( m \) such that the line \( y = mx - 30 \) does not intersect the parabola are \( -4 < m < 28 \).
Answer for screen readers
The values of ( m ) such that the line ( y = mx - 30 ) does not intersect the parabola ( y = 2x^2 + 12x + 2 ) are: $$ -4 < m < 28 $$
Steps to Solve
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Set the equations equal To find when the line does not intersect the parabola, we set the equations equal to each other: $$ mx - 30 = 2x^2 + 12x + 2 $$
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Rearrange into standard form Rearranging gives us: $$ 2x^2 + (12 - m)x + (2 + 30) = 0 $$ This simplifies to: $$ 2x^2 + (12 - m)x + 32 = 0 $$
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Identify the discriminant For the quadratic equation $ax^2 + bx + c = 0$ to have no real solutions, the discriminant must be negative: $$ D = b^2 - 4ac $$ Here, ( a = 2 ), ( b = 12 - m ), and ( c = 32 ). Thus, $$ D = (12 - m)^2 - 4 \cdot 2 \cdot 32 $$
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Set the discriminant less than zero Set the discriminant to be less than zero: $$ (12 - m)^2 - 256 < 0 $$
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Solve the inequality Expanding and solving the inequality: $$ (12 - m)^2 < 256 $$
Taking the square root: $$ |12 - m| < 16 $$
This gives us two inequalities: $$ -16 < 12 - m < 16 $$
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Isolate m Solving these inequalities:
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From ( 12 - m < 16 ): $$ m > -4 $$
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From ( -16 < 12 - m ): $$ m < 28 $$
Thus, the range for ( m ) is: $$ -4 < m < 28 $$
The values of ( m ) such that the line ( y = mx - 30 ) does not intersect the parabola ( y = 2x^2 + 12x + 2 ) are: $$ -4 < m < 28 $$
More Information
This means that for any value of ( m ) within the range of (-4) and (28), the line will never touch the parabola at any point.
Tips
- Forgetting to set the discriminant less than zero when finding conditions for no intersection.
- Miscalculating inequalities during the solving steps, which can lead to incorrect ranges for ( m ).
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