In the xy-plane, a line with equation 2y = 4.5 intersects a parabola at exactly one point. If the parabola has equation y = −4x 2 + bx, where b is a positive constant, what is the value of b?

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Key Explanation: The correct answer is 6. It’s given that a line with equation 2y = 4.5 intersects a parabola with equation y = −4x 2 + bx, where b is a positive constant, at exactly one point in the xy-plane. It follows that the system of equations consisting of 2y = 4.5 and y = −4x 2 + bx has exactly one solution. Dividing both sides of the equation of the line by 2 yields y = 2.25. Substituting 2.25 for y in the equation of the parabola yields 2.25 = −4x 2 + bx. Adding 4x 2 and subtracting bx from both sides of this equation yields 4x 2 − bx + 2.25 = 0. A quadratic equation in the form of ax 2 + bx + c = 0, where a, b, and c are constants, has exactly one solution when the discriminant, b 2 − 4ac, is equal to zero. Substituting 4 for a and 2.25 for c in the expression b 2 − 4ac and setting this expression equal to 0 yields b 2 − 4(4)(2.25) = 0, or b 2 − 36 = 0. Adding 36 to each side of this equation yields b 2 = 36. Taking the square root of each side of this equation yields b = ±6. It’s given that b is positive, so the value of b is 6.

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