Chapter 11

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$9 x^{2}-27 x+18$$

Divide and reduce. Try some by calculator. $$\frac{a^{2}-a-2}{5 a-1} \div \frac{a-2}{10 a^{2}+13 a-3}$$

The formula for the displacement \(s\) of a freely falling body having an initial velocity \(v_{0}\) and acceleration \(a\) is $$s=v_{0} t+\frac{1}{2} a t^{2}$$ Solve this equation for \(a\)

Three masons build \(318 \mathrm{m}\) of wall. Mason \(A\) builds \(7.0 \mathrm{m} /\) day, \(B\) builds 6.0 m/day, and \(C\) builds 5.0 m/day. Mason \(B\) works twice as many days as \(A,\) and \(C\) works half as many days as \(A\) and \(B\) combined. How many days did each work?

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$9 a^{2}-15 a-14$$

An \(8 \frac{3}{4}\) in. length of a certain steel bar weighs \(1 \frac{1}{8}\) ib. Find the weight of a similar bar of length \(22 \frac{2}{3}\) in. in length.

Treat the given numbers in these problems as exact, and leave your answers in fractional form. Do not use your calculator. One crew can assemble \(M\) machines in \(p\) days. Another crew can assemble \(N\) of these machines in \(q\) days. Write an expression for the number of machines that both crews together can assemble in 1 day, combine into a single term, and simplify.

If a carpenter can roof a house in 10 days and another can do the same in 14 days, how many days will it take if they work together?

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$16 c^{2}-48 c+35$$

An amount \(a\) invested at a simple interest rate \(n\) for \(t\) years will accumulate to an amount \(y,\) where \(y=a+a n t .\) Solve for \(a.\)

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$x^{2}+4 x+4$$

The formula for the equivalent resistance \(R\) for the parallel combination of two resistors, \(R_{1}\) and \(R_{2},\) is $$\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}$$ Solve this formula for \(R_{2}.\)

A certain screw machine can produce a box of parts in \(3.3 \mathrm{h}\). A new machine is to be ordered having a speed such that both machines working together would produce a box of parts in 1.4 h. How long would it take the new machine alone to produce a box of parts?

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$x^{2}-30 x+225$$

Treat the given numbers in these problems as exact, and leave your answers in fractional form. Do not use your calculator. If a car travels a distance \(d\) at a constant rate \(V\), the time required will be \(d / V\). The car then continues for a distance \(d_{1}\) at a rate \(V_{1},\) and a third distance \(d_{2}\) at rate \(V_{2} .\) Write an expression for the total travel time; then combine the three terms into a single term and simplify.

A landlord owns a house that consumes 2100 gal of heating oil in three winters. He buys another (insulated) house, and the two houses together use 1850 gal of oil in two winters. How many winters would it take the insulated house alone to use 1250 gal of oil?

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$y^{2}-2 y+1$$

The mass density of an object is its mass divided by its volume. Write and simplify an expression for the density of a sphere having a mass \(m\) and a volume equal to \(4 \pi r^{3} / 3\)

If the resistance of a conductor is \(R_{1}\) at temperature \(t_{1},\) the resistance will change to a value \(R\) when the temperature changes to \(t\), where $$R=R_{1}\left[1+\alpha\left(t-t_{1}\right)\right]$$ and \(\alpha\) is the temperature coefficient of resistance at temperature \(t_{1} .\) Solve this equation for \(t_{1}.\)

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$x^{2}+2 x+1$$

The pressure on a surface is equal to the total force divided by the area. Write and simplify an expression for the pressure on a circular surface of area \(\pi d^{2} / 4\) subjected to a distributed load \(F\)

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$2 y^{2}-12 y+18$$

The stress on a bar in tension is cqual to the load divided by the crosssectional area. Write and simplify an expression for the stress in a bar having a trapezoidal cross section of area \((a+b) h / 2,\) subject to a load \(P\)

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$9-12 a+4 a^{2}$$

The acceleration on a body is equal to the force on the body divided by its mass, and the mass equals the volume of the object times the density. Write and simplify an expression for the acceleration of a sphere having a volume \(4 \pi r^{3} / 3\) and a density \(D,\) subjected to a force \(F\)

Three masses, \(m_{1}, m_{2},\) and \(m_{3},\) are attached together and accelerated by means of a force \(F\), where $$F=m_{1} a+m_{2} a+m_{3} a$$ Solve for the acceleration \(a.\)

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$9+6 x+x^{2}$$

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$4 y^{2}-4 y+1$$

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$9 x^{2}+6 x+1$$

The total energy of a body of mass \(m,\) moving with velocity \(v\) and located at a height \(y\) above some datum, is the sum of the potential energy \(m g y\) and the kinetic energy \(\frac{1}{2} m v^{2} .\) So, $$E=m g y+\frac{1}{2} m v^{2}$$ Solve for \(m.\)

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$16 x^{2}+16 x+4$$

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$9 y^{2}-18 y+9$$

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$16 n^{2}-8 n+1$$

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$16+16 a+4 a^{2}$$

Factor completely, by hand or by calculator. Check your results. The Perfect Square Trinomial. $$1+20 a+100 a^{2}$$

To find two resistors that will give an equivalent resistance of \(400 \Omega\) when wired in series and \(75 \Omega\) when wired in parallel, we must solve the quadratic equation $$R^{2}-400 R+30,000=0$$ Factor the left side of this equation. Factor the left side of this equation.

To find the width \(2 m\) of a road that will give a sight distance of 1000 ft on a curve of radius \(500 \mathrm{ft}\), we must solve the equation $$m^{2}-1000 m+250,000=0$$ Factor the left side of this equation. Factor the left side of this equation.

Project: Some trinomials that have two variables (such as \(x^{2}+5 x y+6 y^{2}\) ) can be factored by temporarily dropping one variable ( \(y\) in this example), factoring the remaining trinomial \(\left(x^{2}+5 x+6\right)\) into \((x+3)(x+2),\) and then putting back the second variable, getting \((x+3 y)(x+2 y) .\) Try this technique on the following trinomials: $$x^{2}-13 x y+36 y^{2} \quad x^{2}+19 x y+84 y^{2} \quad x^{2}-9 x y+20 y^{2}$$