The equation m^(2/3) + 10m^(1/3) + 9 = 0 is said to be in __________form, because making the substitution u = __________results in a new equation that is quadratic.

The equation m^(2/3) + 10m^(1/3) + 9 = 0 is said to be in __________form, because making the substitution u = __________results in a new equation that is quadratic.

Question
Quadratics
asked 2020-10-23
The equation \(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\) is said to be in __________form, because making the substitution u = __________results in a new equation that is quadratic.

Answers (1)

2020-10-24
In mathematical expressions/equations, power to which a variable has been raised is called an exponent or index.
Any indicial equation which can be converted to a quadratic equation using proper substitution can be said to be in “quadratic-form”.
Equation given can be rearranged as:
\(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\)
\(\displaystyle\Leftrightarrow{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}\)
If we make the substitution:
\(\displaystyle{u}={m}^{{\frac{{1}}{{3}}}}\)
The given equation can be transformed into quadratic equation in variable u as shown:
\(\displaystyle{\left({m}^{{\frac{{1}}{{3}}}}\right)}^{{2}}+{10}{\left({m}^{{\frac{{1}}{{3}}}}\right)}+{9}={0}\)
\(\displaystyle\Leftrightarrow{u}^{{2}}+{10}{u}+{9}={0}\)
Answer:
The equation \(\displaystyle{m}^{{\frac{{2}}{{3}}}}+{10}{m}^{{\frac{{1}}{{3}}}}+{9}={0}\) is said to be in quadratic form, because making the substitution \(\displaystyle{u}={m}^{{\frac{{1}}{{3}}}}\) results in a new equation that is quadratic.
0

Relevant Questions

asked 2021-04-25
The unstable nucleus uranium-236 can be regarded as auniformly charged sphere of charge Q=+92e and radius \(\displaystyle{R}={7.4}\times{10}^{{-{15}}}\) m. In nuclear fission, this can divide into twosmaller nuclei, each of 1/2 the charge and 1/2 the voume of theoriginal uranium-236 nucleus. This is one of the reactionsthat occurred n the nuclear weapon that exploded over Hiroshima, Japan in August 1945.
A. Find the radii of the two "daughter" nuclei of charge+46e.
B. In a simple model for the fission process, immediatelyafter the uranium-236 nucleus has undergone fission the "daughter"nuclei are at rest and just touching. Calculate the kineticenergy that each of the "daughter" nuclei will have when they arevery far apart.
C. In this model the sum of the kinetic energies of the two"daughter" nuclei is the energy released by the fission of oneuranium-236 nucleus. Calculate the energy released by thefission of 10.0 kg of uranium-236. The atomic mass ofuranium-236 is 236 u, where 1 u = 1 atomic mass unit \(\displaystyle={1.66}\times{10}^{{-{27}}}\) kg. Express your answer both in joules and in kilotonsof TNT (1 kiloton of TNT releases 4.18 x 10^12 J when itexplodes).
asked 2021-03-09

Show by substitution that \(u(x,t)=\cos(απx)e^−α^2π^2t\) is a solution of the heat equation \(ut=uxx\) on any interval [0, L].

asked 2021-04-19
For the cellar of a new house a hole is dug in the ground, withvertical sides going down 2.40m. A concrete foundation wallis built all the way across the 9.6m width of the excavation. This foundation wall is 0.183m away from the front of the cellarhole. During a rainstorm, drainage from the streetfills up the space in front of the concrete wall, but not thecellar behind the wall. The water does not soak into the clays oil. Find the force the water causes on the foundation wall. For comparison, the weight of the water is given by:
\(\displaystyle{2.40}{m}\cdot{9.60}{m}\cdot{0.183}{m}\cdot{1000}{k}\frac{{g}}{{m}^{{3}}}\cdot{9.8}\frac{{m}}{{s}^{{2}}}={41.3}{k}{N}\)
asked 2021-02-14
Electrocardiographs are often connected as shown in Fig.19-55. The leads are said to be capacitively coupled. A timeconstant of 3.0s is tyical and allows rapid changes in potential tobe recorded accurately. If C = 3.0?F, What value must R have?
asked 2021-05-09
The dominant form of drag experienced by vehicles (bikes, cars,planes, etc.) at operating speeds is called form drag. Itincreases quadratically with velocity (essentially because theamount of air you run into increase with v and so does the amount of force you must exert on each small volume of air). Thus
\(\displaystyle{F}_{{{d}{r}{u}{g}}}={C}_{{d}}{A}{v}^{{2}}\)
where A is the cross-sectional area of the vehicle and \(\displaystyle{C}_{{d}}\) is called the coefficient of drag.
Part A:
Consider a vehicle moving with constant velocity \(\displaystyle\vec{{{v}}}\). Find the power dissipated by form drag.
Express your answer in terms of \(\displaystyle{C}_{{d}},{A},\) and speed v.
Part B:
A certain car has an engine that provides a maximum power \(\displaystyle{P}_{{0}}\). Suppose that the maximum speed of thee car, \(\displaystyle{v}_{{0}}\), is limited by a drag force proportional to the square of the speed (as in the previous part). The car engine is now modified, so that the new power \(\displaystyle{P}_{{1}}\) is 10 percent greater than the original power (\(\displaystyle{P}_{{1}}={110}\%{P}_{{0}}\)).
Assume the following:
The top speed is limited by air drag.
The magnitude of the force of air drag at these speeds is proportional to the square of the speed.
By what percentage, \(\displaystyle{\frac{{{v}_{{1}}-{v}_{{0}}}}{{{v}_{{0}}}}}\), is the top speed of the car increased?
Express the percent increase in top speed numerically to two significant figures.
asked 2021-05-05
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
asked 2021-05-16
Look Out! A snowball rolls off a barn roof that slopes downward at an angle of 40 degrees . The edge of the roof is 14.0 m above the ground, and the snowball has a speed of 7.00 m/s as it rolls off the roof. Ignore air resistance.
A man 1.9 m tall is standing 4.0 m from the edge of the barn. Will he be hit by the snowball?
asked 2021-05-06
a. Draw all ofthe possible isomers for diacetylferrocene which could be producedby the acetylation of ferrocene.
b. Of these isomers, which is most likely to be formed in thereaction and why?
c. Give the structure of the electrophilic species which actuallyattacks ferrocene in the substitution reaction and the equation forhow it is formed by the reaction of acetic anhydride withphosphoric acid.
asked 2021-02-23
Solid NaBr is slowly added to a solution that is 0.010 M inCu+ and 0.010 M in Ag+. (a) Which compoundwill begin to precipitate first? (b) Calculate [Ag+] when CuBr justbegins to precipitate. (c) What percent of Ag+ remains in solutionat this point?
a) AgBr: \(\displaystyle{\left({0.010}+{s}\right)}{s}={4.2}\cdot{10}^{{-{8}}}\) \(\displaystyle{s}={4.2}\cdot{10}^{{-{9}}}{M}{B}{r}\) needed form PPT
CuBr: \(\displaystyle{\left({0.010}+{s}\right)}{s}={7.7}\cdot{\left({0.010}+{s}\right)}{s}={7.7}\cdot{10}^{{-{13}}}\) Ag+=\(\displaystyle{1.8}\cdot{10}^{{-{7}}}\)
b) \(\displaystyle{4.2}\cdot{10}^{{-{6}}}{\left[{A}{g}+\right]}={7.7}\cdot{10}^{{-{13}}}\) [Ag+]\(\displaystyle={1.8}\cdot{10}^{{-{7}}}\)
c) \(\displaystyle{\frac{{{1.8}\cdot{10}^{{-{7}}}}}{{{0.010}{M}}}}\cdot{100}\%={0.18}\%\)
asked 2021-02-19
An airplane propeller is 2.08 m in length (from tip to tip) and has a mass of 117 kg. When the airpline's engine is first started, it applies a constant torque of \(\displaystyle{1950}\ {N}\cdot{m}\) to the propeller, which starts from rest.
a) What is the angular acceleration of the propeller? Model the propeller as a slender rod.
b) What is the propeller's angular speed after making 5.00 revolutions?
c) How much work is done by the engine during the first 5.00 revolutions?
e) What is the instantaneous power output of the motor at the instant that the propeller has turne through 5.00 revolutions?
...