Chapter 7

Solve the given quadratic equations by completing the square. $$\pi^{2} y^{2}+2 \pi y=3$$

Use a calculator to graph all three parabolas on the same coordinate system. . Describe (a) the shifts and (b) the stretching and shrinking. (a) \(y=x^{2}\) (b) \(y=3 x^{2}\) (c) \(y=\frac{1}{3} x^{2}\)

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$0.29 Z^{2}-0.18=0.63 Z$$

Solve the given quadratic equations by factoring. $$4 x(x+1)=3$$

Solve the given quadratic equations by completing the square. $$9 x^{2}+6 x+1=0$$

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$13.2 x=15.5-12.5 x^{2}$$

Solve the given quadratic equations by factoring. $$t(43+t)=9-9 t^{2}$$

Solve the given applied problem. Use a calculator to find the vertex of \(s=-9.8 t^{2}+25 t+4\). Round the coordinates to the nearest hundredth.

Solve the given applied problem. Find the range of the function \(s=-16 t^{2}+64 t+6\).

Solve the given quadratic equations by factoring. $$2 x^{2}-7 a x+4 a^{2}=a^{2}$$

Solve the given applied problem. Find the equation of the quadratic function that has vertex (0,0) and passes through the point (25,125).

Solve the given quadratic equations by factoring. $$8 s^{2}+16 s=90$$

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$b^{2} x^{2}+1-a=(b+1) x$$

Use completing the square to solve the given problems. The voltage \(V\) across a certain electronic device is related to the temperature \(T\) (in \(^{\circ}\) C) by \(V=4.0 T-0.2 T^{2}\). For what temperature(s) is \(V=15 \mathrm{V} ?\)

Solve the given applied problem. A parabolic satellite dish is 8.40 in. deep and 36.0 in. across its opening. If the dish is positioned so it opens directly upward with its vertex at the origin, find the equation of its parabolic cross section.

Solve the given quadratic equations by factoring. $$18 t^{2}=48 t-32$$

Solve the given quadratic equations using the quadratic formula. If there are no real roots, state this as the answer. Exercises \(3-6\) are the same as Exercises \(13-16\) of Section 7.2. $$c^{2} x^{2}-x-1=x^{2}$$

Use completing the square to solve the given problems. A flare is shot vertically into the air such that its distance \(s\) (in \(\mathrm{ft}\) ) above the ground is given by \(s=64 t-16 t^{2},\) where \(t\) is the time (in s) after it was fired. Find \(t\) for \(s=48 \mathrm{ft}\).

Solve the given applied problem. Find the smallest integer value of \(c\) such that \(y=2 x^{2}-4 x-c\) has at least one real root.

Without solving the given equations, determine the character of the roots. $$2 x^{2}-7 x=-8$$

Solve the given quadratic equations by factoring. $$(x+2)^{3}=x^{3}+8$$

Use completing the square to solve the given problems. A woman is holding a selfie stick so her cell phone camera is exactly 30 in. from her face. The horizontal distance between the woman's face and the cell phone is exactly 6 in. more than the vertical distance. How far above her face is the cell phone?

Solve the given applied problem. Find the smallest integer value of \(c\) such that \(y=3 x^{2}-12 x+c\) has no real roots.

Without solving the given equations, determine the character of the roots. $$3 x^{2}=14-19 x$$

Solve the given quadratic equations by factoring. $$V\left(V^{2}-4\right)=V^{2}(V-1)$$

Use completing the square to solve the given problems. A rectangular storage area is \(8.0 \mathrm{m}\) longer than it is wide. If the area is \(28 \mathrm{m}^{2},\) what are its dimensions?

Solve the given applied problem. Find the equation of the parabola that contains the points \((-2,-3),(0,-3),\) and \((2,5)\).

Without solving the given equations, determine the character of the roots. $$3.6 t^{2}+2.1=7.7 t$$

Solve the given quadratic equations by factoring. $$(x+a)^{2}-b^{2}=0$$

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. $$b x^{2}-b=x-b^{2} x$$

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

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

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 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. 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.

Solve the given problems. All numbers are accurate to at least two significant digits. Without drawing the graph or completely solving the equation, explain how to find the number of \(x\) -intercepts of a quadratic function.

Solve the given applied problem. A computer analysis of the power \(P\) (in W) used by a pressing machine shows that \(P=50 i-3 i^{2},\) where \(i\) is the current (in A). Sketch the graph of \(P=f(i)\).

Solve the given problems. All numbers are accurate to at least two significant digits. Use the discriminant \(b^{2}-4 a c\) to determine if the equation \(90 x^{2}-123 x+40=0\) can be solved by factoring. Explain why or why not. Do not solve.

In finding the dimensions of a crate, the equation \(12 x^{2}-64 x+64=0\) is used. Solve for \(x,\) if \(x>2\).

Solve the given applied problem. Tests show that the power \(P\) (in \(\mathrm{hp}\) ) of an automobile engine as a function of \(r\) (in \(\mathrm{r} / \mathrm{min}\) ) is given by \(P=-5.0 \times 10^{-6} r^{2}+0.050 r-45(1500

Solve the given problems. All numbers are accurate to at least two significant digits. Solve \(6 x^{2}-x=15\) for \(x\) by (a) factoring, (b) completing the square, and (c) the quadratic formula. Which is (a) longest? (b) shortest?

If a rocket is launched with an initial velocity of \(320 \mathrm{ft} / \mathrm{s}\), its height above ground after \(t\) seconds is given by \(-16 t^{2}+320 t\) (in ft). Find the times when the height is \(0 .\)

Solve the given applied problem. The height \(h\) (in \(\mathrm{m}\) ) of a fireworks shell shot vertically upward as a function of time \(t\) (in s) is \(h=-4.9 t^{2}+68 t+2 .\) How long should the fuse last so that the shell explodes at the top of its trajectory?

Solve the given problems. All numbers are accurate to at least two significant digits. In machine design, in finding the outside diameter \(D_{0}\) of a hollow shaft, the equation \(D_{0}^{2}-D D_{0}-0.25 D^{2}=0\) is used. Solve for \(D_{0}\) if \(D=3.625 \mathrm{cm}\).

The voltage \(V\) across a semiconductor in a computer is given by \(V=\alpha I+\beta I^{2},\) where \(I\) is the current (in A). If a 6 -V battery is conducted across the semiconductor, find the current if \(\alpha=2 \Omega\) and \(\beta=0.5 \Omega / \mathrm{A}\).