Tutors Answer Your Questions about Quadratic Equations (FREE)
Question 1172098: So for this question(All the roots of
x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a), I did these steps:
all roots of quadratic equation ax^2+bx+c =0 are real if and only if
discriminant D = b^2- 4ac >= 0
I know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0
some simplification of second equation:
x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q
So we want to prove that discriminant of equation
3x^2+x(2p + 2a) + ap + q = 0
is greater or equal to zero.
.
D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q)
To prove that D >=0 , we can view D as polynomial of a:
.
D(a) = 4a^2 - a(4p) + 4(p^2-3q) if quadratic polynomial have positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive.
So remains to prove that
(4p)^2 - 4*4*4(p^2-3q) <= 0
let's divide both side by 4^2 :
p^2 - 4(p^2-3q) <= 0
-3p^2 + 12q <= 0
divide by 3
-p^2+4q<= 0
transfer terms to the other side:
0<= p^2 -4q
Are these steps correct? Would there be another approach to solving this? If there is, what is it? Thanks.
Click here to see answer by ikleyn(52776)   |
Question 1172144: For this problem: All the roots of
x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a.
I solved it like this:
We know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0
Let us do some simplification of second equation:
x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q
So we want to prove that the discriminant of equation
3x^2+x(2p + 2a) + ap + q = 0
is greater or equal to zero.
D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q)
To prove that D >=0 , we can view D as a polynomial of a:
D(a) = 4a^2 - a(4p) + 4(p^2-3q)
We know that if a quadratic polynomial has a positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive.
So it remains for us to prove that
(4p)^2 - 4*4*4(p^2-3q) <= 0
Let's divide both side by 4^2 :
p^2 - 4(p^2-3q) <= 0
-3p^2 + 12q <= 0
Now we divide by 3
-p^2+4q<= 0
Now we transfer terms to the other side:
0<= p^2 -4q, or p^2-4q >=0
For this problem, is there any other ways to solve it? If so, can you show me? Thanks!
Click here to see answer by Edwin McCravy(20054)  |
Question 1172141: For this problem: All the roots of
x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a.
I solved it like this:
We know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0
Let us do some simplification of second equation:
x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q
So we want to prove that the discriminant of equation
3x^2+x(2p + 2a) + ap + q = 0
is greater or equal to zero.
D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q)
To prove that D >=0 , we can view D as a polynomial of a:
D(a) = 4a^2 - a(4p) + 4(p^2-3q)
We know that if a quadratic polynomial has a positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive.
So it remains for us to prove that
(4p)^2 - 4*4*4(p^2-3q) <= 0
Let's divide both side by 4^2 :
p^2 - 4(p^2-3q) <= 0
-3p^2 + 12q <= 0
Now we divide by 3
-p^2+4q<= 0
Now we transfer terms to the other side:
0<= p^2 -4q, or p^2-4q >=0
There, now we have proved that the roots of a are real!
Can you show me another way to solve it? I'm curious.
Click here to see answer by Edwin McCravy(20054) |
Question 1172141: For this problem: All the roots of
x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a.
I solved it like this:
We know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0
Let us do some simplification of second equation:
x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q
So we want to prove that the discriminant of equation
3x^2+x(2p + 2a) + ap + q = 0
is greater or equal to zero.
D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q)
To prove that D >=0 , we can view D as a polynomial of a:
D(a) = 4a^2 - a(4p) + 4(p^2-3q)
We know that if a quadratic polynomial has a positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive.
So it remains for us to prove that
(4p)^2 - 4*4*4(p^2-3q) <= 0
Let's divide both side by 4^2 :
p^2 - 4(p^2-3q) <= 0
-3p^2 + 12q <= 0
Now we divide by 3
-p^2+4q<= 0
Now we transfer terms to the other side:
0<= p^2 -4q, or p^2-4q >=0
There, now we have proved that the roots of a are real!
Can you show me another way to solve it? I'm curious.
Click here to see answer by AnlytcPhil(1806)  |
Question 1172141: For this problem: All the roots of
x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a.
I solved it like this:
We know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0
Let us do some simplification of second equation:
x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q
So we want to prove that the discriminant of equation
3x^2+x(2p + 2a) + ap + q = 0
is greater or equal to zero.
D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q)
To prove that D >=0 , we can view D as a polynomial of a:
D(a) = 4a^2 - a(4p) + 4(p^2-3q)
We know that if a quadratic polynomial has a positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive.
So it remains for us to prove that
(4p)^2 - 4*4*4(p^2-3q) <= 0
Let's divide both side by 4^2 :
p^2 - 4(p^2-3q) <= 0
-3p^2 + 12q <= 0
Now we divide by 3
-p^2+4q<= 0
Now we transfer terms to the other side:
0<= p^2 -4q, or p^2-4q >=0
There, now we have proved that the roots of a are real!
Can you show me another way to solve it? I'm curious.
Click here to see answer by Plocharczyk(17) |
Question 1172141: For this problem: All the roots of
x^2 + px + q = 0 are real, where p and q are real numbers. Prove that all the roots of
x^2 + px + q + (x + a)(2x + p) = 0 are real, for any real number a.
I solved it like this:
We know that all roots of x^2+px+q=0 are real. We can derive from this condition that p^2- 4q >=0
Let us do some simplification of second equation:
x^2+ px + q (x+a)(2x+p) = 3x^2+ x(2p + 2a) + ap + q
So we want to prove that the discriminant of equation
3x^2+x(2p + 2a) + ap + q = 0
is greater or equal to zero.
D = (2p + 2a)^2 - 4 *3(ap+q) = 4(p^2 + a^2 + 2ap - 3ap - 3q) = 4(a^2-ap+p^2-3q)
To prove that D >=0 , we can view D as a polynomial of a:
D(a) = 4a^2 - a(4p) + 4(p^2-3q)
We know that if a quadratic polynomial has a positive greatest coefficient and it's discriminant <= 0 then polynomials are always positive.
So it remains for us to prove that
(4p)^2 - 4*4*4(p^2-3q) <= 0
Let's divide both side by 4^2 :
p^2 - 4(p^2-3q) <= 0
-3p^2 + 12q <= 0
Now we divide by 3
-p^2+4q<= 0
Now we transfer terms to the other side:
0<= p^2 -4q, or p^2-4q >=0
There, now we have proved that the roots of a are real!
Can you show me another way to solve it? I'm curious.
Click here to see answer by mccravyedwin(406)  |
Question 1172541: The weekly profit of your group’s home-made brownies in a box is modeled by the equation profit, P = - x2 + 120x - 28. The weekly profit P is dependent on the number of boxes of brownies x sold. If the break-even point is when P = 0, then how many boxes of brownies must your group sell in a week in order to break-even your profit?
Click here to see answer by Solver92311(821)  |
Question 1172677: The profit on t-shirts sold by a school is determined by the quadratic equation P=-x^2+30x+400 where x is the number of T-shirts sold at $10 each and P is the profit in dollars.
a) determine the number of tshirts that must be sold to break even?
Click here to see answer by ikleyn(52776) |
Question 1172916: The profit, P, in dollars, of an amusement park is modelled by P = −25x^2 + 950x + 3000, where x is the
ticket price in dollars. Demonstrate the knowledge of completing the square to determine the maximum profit
and the charge per ticket to make the maximum profit. What ticket price(s) will generate a profit of $7125?
pls i need help
Click here to see answer by ikleyn(52776) |
Question 1172916: The profit, P, in dollars, of an amusement park is modelled by P = −25x^2 + 950x + 3000, where x is the
ticket price in dollars. Demonstrate the knowledge of completing the square to determine the maximum profit
and the charge per ticket to make the maximum profit. What ticket price(s) will generate a profit of $7125?
pls i need help
Click here to see answer by josgarithmetic(39616) |
Question 1173107: Please help me solve this quadratic equation and round the answer to the nearest hundredth
I also need to know the correct "form." I'm new to quadratic equations and am struggling
form options
(x+[])^2=[] or (x-[])^2=[]
equation:
x^2+6x+7=0
Click here to see answer by Boreal(15235)  |
Question 1173107: Please help me solve this quadratic equation and round the answer to the nearest hundredth
I also need to know the correct "form." I'm new to quadratic equations and am struggling
form options
(x+[])^2=[] or (x-[])^2=[]
equation:
x^2+6x+7=0
Click here to see answer by Edwin McCravy(20054) |
Question 1173107: Please help me solve this quadratic equation and round the answer to the nearest hundredth
I also need to know the correct "form." I'm new to quadratic equations and am struggling
form options
(x+[])^2=[] or (x-[])^2=[]
equation:
x^2+6x+7=0
Click here to see answer by MathTherapy(10551)  |
Question 1173107: Please help me solve this quadratic equation and round the answer to the nearest hundredth
I also need to know the correct "form." I'm new to quadratic equations and am struggling
form options
(x+[])^2=[] or (x-[])^2=[]
equation:
x^2+6x+7=0
Click here to see answer by ikleyn(52776) |
Question 1173126: An object is thrown vertically upward with a velocity of 96m/sec. The distance S(t) above the ground after t second is given by the formula S(t)= 96t-5t²
a. How high will it be at the end of 3 seconds?
b. How much time will it take the object to be 172 m above the ground?
Click here to see answer by ikleyn(52776) |
Question 1173134: help me please
A ball is thrown from a height of 45 meters with an initial downward velocity of 5 m/s. The ball's height h (in meters) after t seconds is given by the following.
h=45-5t-5t^2
How long after the ball is thrown does it hit the ground?
Round your answer(s) to the nearest hundredth.
Click here to see answer by ewatrrr(24785)  |
Question 1173136: please help
A ball is thrown from an initial height of 5 feet with an initial upward velocity of 21 ft/s. The ball's height h (in feet) after t seconds is given by the following.
h=5+21t-16t^2
Find all values of for which the ball's height is feet.
Round your answer(s) to the nearest hundredth.
Click here to see answer by ikleyn(52776) |
Question 1172917: Officer Mark has 300 m of yellow tape to seal off the area of a crime scene. Using the tape, he must divide the area
into four identical rectangular sections, side by side. What dimensions of the crime scene will maximize the area?
What is the maximum area that can be enclosed?
pls i need the help now its really difficult for me thank u
Click here to see answer by Boreal(15235) |
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7606..7650, 7651..7695, 7696..7740, 7741..7785, 7786..7830, 7831..7875, 7876..7920, 7921..7965, 7966..8010, 8011..8055, 8056..8100, 8101..8145, 8146..8190, 8191..8235, 8236..8280, 8281..8325, 8326..8370, 8371..8415, 8416..8460, 8461..8505, 8506..8550, 8551..8595, 8596..8640, 8641..8685, 8686..8730, 8731..8775, 8776..8820, 8821..8865, 8866..8910, 8911..8955, 8956..9000, 9001..9045, 9046..9090, 9091..9135, 9136..9180, 9181..9225, 9226..9270, 9271..9315, 9316..9360, 9361..9405, 9406..9450, 9451..9495, 9496..9540, 9541..9585, 9586..9630, 9631..9675, 9676..9720, 9721..9765, 9766..9810, 9811..9855, 9856..9900, 9901..9945, 9946..9990, 9991..10035, 10036..10080, 10081..10125, 10126..10170, 10171..10215, 10216..10260, 10261..10305, 10306..10350, 10351..10395, 10396..10440, 10441..10485, 10486..10530, 10531..10575, 10576..10620, 10621..10665, 10666..10710, 10711..10755, 10756..10800, 10801..10845, 10846..10890, 10891..10935, 10936..10980, 10981..11025, 11026..11070, 11071..11115, 11116..11160, 11161..11205, 11206..11250, 11251..11295, 11296..11340, 11341..11385, 11386..11430, 11431..11475, 11476..11520, 11521..11565, 11566..11610, 11611..11655, 11656..11700, 11701..11745, 11746..11790, 11791..11835, 11836..11880, 11881..11925, 11926..11970, 11971..12015, 12016..12060, 12061..12105, 12106..12150, 12151..12195, 12196..12240, 12241..12285, 12286..12330, 12331..12375, 12376..12420, 12421..12465, 12466..12510, 12511..12555, 12556..12600, 12601..12645, 12646..12690, 12691..12735, 12736..12780, 12781..12825, 12826..12870, 12871..12915, 12916..12960, 12961..13005, 13006..13050, 13051..13095, 13096..13140, 13141..13185, 13186..13230, 13231..13275, 13276..13320, 13321..13365, 13366..13410, 13411..13455, 13456..13500, 13501..13545, 13546..13590, 13591..13635, 13636..13680, 13681..13725, 13726..13770, 13771..13815, 13816..13860, 13861..13905, 13906..13950, 13951..13995, 13996..14040, 14041..14085, 14086..14130, 14131..14175, 14176..14220, 14221..14265, 14266..14310, 14311..14355, 14356..14400, 14401..14445, 14446..14490, 14491..14535, 14536..14580, 14581..14625, 14626..14670, 14671..14715, 14716..14760, 14761..14805, 14806..14850, 14851..14895, 14896..14940, 14941..14985, 14986..15030, 15031..15075, 15076..15120, 15121..15165, 15166..15210, 15211..15255, 15256..15300, 15301..15345, 15346..15390, 15391..15435, 15436..15480, 15481..15525, 15526..15570, 15571..15615, 15616..15660, 15661..15705, 15706..15750, 15751..15795, 15796..15840, 15841..15885, 15886..15930, 15931..15975, 15976..16020, 16021..16065, 16066..16110, 16111..16155, 16156..16200, 16201..16245, 16246..16290, 16291..16335, 16336..16380, 16381..16425, 16426..16470, 16471..16515, 16516..16560, 16561..16605, 16606..16650, 16651..16695, 16696..16740, 16741..16785, 16786..16830, 16831..16875, 16876..16920, 16921..16965, 16966..17010, 17011..17055, 17056..17100, 17101..17145, 17146..17190, 17191..17235, 17236..17280, 17281..17325, 17326..17370, 17371..17415, 17416..17460, 17461..17505, 17506..17550, 17551..17595, 17596..17640, 17641..17685, 17686..17730, 17731..17775, 17776..17820, 17821..17865, 17866..17910, 17911..17955, 17956..18000, 18001..18045, 18046..18090, 18091..18135, 18136..18180, 18181..18225, 18226..18270, 18271..18315, 18316..18360, 18361..18405, 18406..18450, 18451..18495, 18496..18540, 18541..18585, 18586..18630, 18631..18675, 18676..18720, 18721..18765, 18766..18810, 18811..18855, 18856..18900, 18901..18945, 18946..18990, 18991..19035, 19036..19080, 19081..19125, 19126..19170, 19171..19215, 19216..19260, 19261..19305, 19306..19350, 19351..19395, 19396..19440, 19441..19485, 19486..19530, 19531..19575, 19576..19620, 19621..19665, 19666..19710, 19711..19755, 19756..19800, 19801..19845, 19846..19890, 19891..19935, 19936..19980, 19981..20025, 20026..20070, 20071..20115, 20116..20160, 20161..20205, 20206..20250, 20251..20295, 20296..20340, 20341..20385, 20386..20430, 20431..20475, 20476..20520, 20521..20565, 20566..20610, 20611..20655, 20656..20700, 20701..20745, 20746..20790, 20791..20835, 20836..20880, 20881..20925, 20926..20970, 20971..21015, 21016..21060, 21061..21105, 21106..21150, 21151..21195, 21196..21240, 21241..21285, 21286..21330, 21331..21375, 21376..21420, 21421..21465, 21466..21510, 21511..21555, 21556..21600, 21601..21645, 21646..21690, 21691..21735, 21736..21780, 21781..21825, 21826..21870, 21871..21915, 21916..21960, 21961..22005, 22006..22050, 22051..22095, 22096..22140, 22141..22185, 22186..22230, 22231..22275, 22276..22320, 22321..22365, 22366..22410, 22411..22455, 22456..22500, 22501..22545, 22546..22590, 22591..22635, 22636..22680, 22681..22725, 22726..22770, 22771..22815, 22816..22860, 22861..22905, 22906..22950, 22951..22995, 22996..23040, 23041..23085, 23086..23130, 23131..23175, 23176..23220, 23221..23265, 23266..23310, 23311..23355, 23356..23400, 23401..23445, 23446..23490, 23491..23535, 23536..23580, 23581..23625, 23626..23670, 23671..23715, 23716..23760, 23761..23805, 23806..23850, 23851..23895, 23896..23940, 23941..23985, 23986..24030, 24031..24075, 24076..24120, 24121..24165, 24166..24210, 24211..24255, 24256..24300, 24301..24345, 24346..24390, 24391..24435, 24436..24480, 24481..24525, 24526..24570, 24571..24615, 24616..24660, 24661..24705, 24706..24750, 24751..24795, 24796..24840, 24841..24885, 24886..24930, 24931..24975, 24976..25020, 25021..25065, 25066..25110, 25111..25155, 25156..25200, 25201..25245, 25246..25290, 25291..25335, 25336..25380, 25381..25425, 25426..25470, 25471..25515, 25516..25560, 25561..25605, 25606..25650, 25651..25695, 25696..25740, 25741..25785, 25786..25830, 25831..25875, 25876..25920, 25921..25965, 25966..26010, 26011..26055, 26056..26100, 26101..26145, 26146..26190, 26191..26235, 26236..26280, 26281..26325, 26326..26370, 26371..26415, 26416..26460, 26461..26505, 26506..26550, 26551..26595, 26596..26640, 26641..26685, 26686..26730, 26731..26775, 26776..26820, 26821..26865, 26866..26910, 26911..26955, 26956..27000, 27001..27045, 27046..27090, 27091..27135, 27136..27180, 27181..27225, 27226..27270, 27271..27315, 27316..27360, 27361..27405, 27406..27450, 27451..27495, 27496..27540, 27541..27585, 27586..27630, 27631..27675, 27676..27720, 27721..27765, 27766..27810, 27811..27855, 27856..27900, 27901..27945, 27946..27990, 27991..28035, 28036..28080, 28081..28125, 28126..28170, 28171..28215, 28216..28260, 28261..28305, 28306..28350, 28351..28395, 28396..28440, 28441..28485, 28486..28530, 28531..28575, 28576..28620, 28621..28665, 28666..28710, 28711..28755, 28756..28800, 28801..28845, 28846..28890, 28891..28935, 28936..28980, 28981..29025, 29026..29070, 29071..29115, 29116..29160, 29161..29205, 29206..29250, 29251..29295, 29296..29340, 29341..29385, 29386..29430, 29431..29475, 29476..29520, 29521..29565, 29566..29610, 29611..29655, 29656..29700, 29701..29745, 29746..29790, 29791..29835, 29836..29880, 29881..29925, 29926..29970, 29971..30015, 30016..30060, 30061..30105, 30106..30150, 30151..30195, 30196..30240
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