91Ó°ÊÓ

Solve the given applied problem. Find the equation of the parabola that contains the points \((-1,14)(1,9),\) and \((2,8)\).

Solve the given quadratic equations by factoring. $$x^{2}+2 a x=b^{2}-a^{2}$$

Solve the given problems. All numbers are accurate to at least two significant digits. Find \(k\) if the equation \(x^{2}+4 x+k=0\) has a real double root.

Solve the given applied problem. The vertical distance \(d\) (in \(\mathrm{cm}\) ) of the end of a robot arm above a conveyor belt in its 8 -s cycle is given by \(d=2 t^{2}-16 t+47\). Sketch the graph of \(d=f(t)\).

Solve the given quadratic equations by factoring. $$x^{2}\left(a^{2}+2 a b+b^{2}\right)=x(a+b)$$

Solve the given applied problem. When mineral deposits form a uniform coating \(1 \mathrm{mm}\) thick on the inside of a pipe of radius \(r\) (in \(m m\) ), the cross-sectional area \(A\) through which water can flow is \(A=\pi\left(r^{2}-2 r+1\right) .\) Sketch \(A=f(r)\).

Solve the given problems. All numbers are accurate to at least two significant digits. Find the smallest positive integer value of \(k\) if the equation \(x^{2}+3 x+k=0\) has roots with imaginary numbers.

Solve the given problems. All numbers are accurate to at least two significant digits. Solve the equation \(x^{4}-5 x^{2}+4=0\) for \(x\). [Hint: The equation can be written as \(\left.\left(x^{2}\right)^{2}-5\left(x^{2}\right)+4=0 . \text { First solve for } x^{2} .\right]\)

Solve the given applied problem. The shape of the Gateway Arch in St. Louis can be approximated by the parabola \(y=192-0.0208 x^{2}\) (in meters) if the origin is at ground level, under the center of the Arch. Display the equation representing the Arch on a calculator. How high and wide is the Arch?

Solve the given applied problem. Under specified conditions, the pressure loss \(L\) (in Ib/in. \(^{2}\) per \(100 \mathrm{ft}),\) in the flow of water through a fire hose in which the flow is \(q\) gal/min, is given by \(L=0.0002 q^{2}+0.005 q\). Sketch the graph of \(L\) as a function of \(q,\) for \(q<100\) gal/min.