state the discriminant of the quadratic formula and describe for which values of the discriminant a quadratic equation will have real solutions, for which values it will have imaginary solutions, and for which values it will have one solution

List all possible rational zeros given by the Rational Zeros Theorem (but don't check to see which actually are zeros). (Enter your answers as a comma-separated list.) Q(x) = x4 − 5x3 − 3x + 6

Find the coordinates of the vertex for the parabola defined by the given quadratic function.

f(x) = x2 + 8x - 8

A bottle of white wine at room temperature (68°F) is placed in a refrigerator at 4 p.m. Its temperature after t hr is changing at the rate of −18e^−0.6t °F/hr. By how many degrees will the temperature of the wine have dropped by 10 p.m.? (Round your answer to one decimal place.)

What will the temperature of the wine be at 10 p.m.? (Round your answer to one decimal place.)

Vector Analysis: Verify Green’s Theorem in the plane for ? ⃑ = (?^2 + ?^2)?̂+ (?^2 − ?^2)?̂ in the anti-clockwise direction around the ellipse 4?^2 + ?^2 = 16.

A boat leaves the harbor entrance and travels 27 miles in the direction ??? ° ?. The captain then travels for another 16 miles in the direction ??? ° ?, which is the boat’s final position. How far is the harbor entrance from the boat’s final position? What is the bearing of the boat from the harbor entrance? Round your answers to the nearest WHOLE numbers.

y''-2y'+y=e^xx^-1

Suppose a rancher wants to fence off a rectangular shaped enclosure. He has 2400 feet of fencing that he can use. What are the dimensions that gives the enclosure the most area?

(a) Draw a sketch of the pen in this question. Appropriately label the relevant information in your sketch.

(b) Based on your sketch above, what equation is being maximized?

(c) Based on your sketch above, what equation represents the given constraint?

(d) Find the dimensions of the enclosure that gives the largest area.

(e) How much fence is used on the East-West sides? How much fence is used on the North-South sides? What is the ration of the amount of fence used on the East-West sides to the amount of fence used on the North-South sides?

Let P denote the vector space of all polynomials with real coefficients and P_n be the set of all polynomials in p with degree <= n.

a) Show that P_n is a vector subspace of P.

b) Show that {1,x,x²,...,xⁿ} is a basis for P_n.

Solve the following equation using the method of variation of parameters :

x² y'' − 2y = 3x² − 1, x > 0.

Find the local extrema of the following function.

f(x,y)=108+7xy-5x²-6y²

Question: (a) Find parametric equations for the line of intersection of the planes given by 3x − 2y + z = 1 and 2x + y − 3z = 3.
(b) Find the equation of the plane orthogonal to both of these planes and passing through the point (−2, 1, 1).

({s^3} + 6{s^2} + 15s + 15)=0

Solve for s - please show all work