Square Root Calculator
Square Root Calculator
About This Tool
The online Square Root Calculator is used to find the square root of the number you enter.
Square Root
In mathematics, a square root of a number x is a number r such that r2 = x.
For example,
1. The square root of 25 is 5 because 52 = 25.
3. The square root of 2 is 1. approximately.
3. The square root of Pi (&) is 1. approximately.
Square Root Table
The following is the square root table from 1 to 1000 rounded to 5 digits:
x√x1121.4142131.732054252.2360762.4494972.6457582.8284393103.16228113.31662123.4641133.60555143.74166153.87298164174.12311184.24264194.3589204.47214214.58258224.69042234.79583244.89898255265.09902275.19615285.2915295.38516305.47723315.56776325.65685335.74456345.83095355.91608366376.08276386.16441396.245406.32456416.40312426.48074436.55744446.63325456.7082466.78233476.85565486.9282497507.07107517.14143527.2111537.28011547.34847557.4162567.48331577.54983587.61577597.68115607.74597617.81025627.87401637.93725648658.06226668.12404678.18535688.24621698.30662708.3666718.42615728.48528738.544748.60233758.66025768.7178778.77496788.83176798.88819808.94427819829.05539839.11043849.16515859.21954869.27362879.32738889.38083899.43398909.48683919.53939929.59166939.64365949.69536959.74679969.79796979.84886989.89949999.949871001010110.0498810210.099510310.1488910410.1980410510.2469510610.2956310710.3440810810.392310910.4403111010.4880911110.5356511210.5830111310.6301511410.6770811510.7238111610.7703311710.8166511810.8627811910.9087112010.954451211112211.0453612311.0905412411.1355312511.1803412611.2249712711.2694312811.3137112911.3578213011.4017513111.4455213211.4891313311.5325613411.5758413511.6189513611.661913711.704713811.7473413911.7898314011.8321614111.8743414211.9163814311.958261441214512.0415914612.0830514712.1243614812.1655314912.2065615012.2474515112.2882115212.3288315312.3693215412.4096715512.449915612.4915712.5299615812.5698115912.6095216012.6491116112.6885816212.7279216312.7671516412.8062516512.8452316612.884116712.9228516812.961481691317013.038417113.076717213.1148817313.1529517413.1909117513.2287617613.266517713.3041317813.3416617913.3790918013.4164118113.4536218213.4907418313.5277518413.5646618513.6014718613.6381818713.6747918813.7113118913.7477319013.7840519113.8202719213.8564119313.8924419413.9283919513.964241961419714.0356719814.0712519914.1067420014.1421420114.1774520214.2126720314.2478120414.2828620514.3178220614.352720714.3874920814.4222120914.4568321014.4913821114.5258421214.5602221314.5945221414.6287421514.6628821614.6969421714.7309221814.7648221914.7986522014.832422114.8660722214.8996622314.9331822414.966632251522615.033322715.0665222815.0996722915.1327523015.1657523115.1986823215.2315523315.2643423415.2970623515.3297123615.3622923715.394823815.4272523915.4596224015.4919324115.5241724215.5563524315.5884624415.620524515.6524824615.6843924715.7162324815.7480224915.7797325015.8113925115.8429825215.8745125315.9059725415.9373825515.968722561625716.0312225816.0623825916.0934826016.1245226116.1554926216.1864126316.2172726416.2480826516.2788226616.3095126716.3401326816.3707126916.4012227016.4316827116.4620827216.4924227316.5227127416.5529527516.5831227616.6132527716.6433227816.6733327916.7032928016.733228116.7630528216.7928628316.822628416.852328516.8819428616.9115328716.9410728816.970562891729017.0293929117.0587229217.0880129317.1172429417.1464329517.1755629617.2046529717.2336929817.2626829917.2916230017.3205130117.3493530217.3781530317.406930417.435630517.4642530617.4928630717.5214230817.5499330917.578431017.6068231117.6351931217.6635231317.6918131417.7200531517.7482431617.7763931717.8044931817.8325531917.8605732017.8885432117.9164732217.9443632317.97223241832518.0277632618.0554732718.0831432818.1107732918.1383633018.165933118.1934133218.2208733318.2482933418.2756733518.3030133618.330333718.3575633818.3847833918.4119534018.4390934118.4661934218.4932434318.5202634418.5472434518.5741834618.6010834718.6279434818.6547634918.6815435018.7082935118.7349935218.7616635318.7882935418.8148935518.8414435618.8679635718.8944435818.9208935918.947336018.973673611936219.026336319.0525636419.0787836519.1049736619.1311336719.1572436819.1833336919.2093737019.2353837119.2613637219.287337319.3132137419.3390837519.3649237619.3907237719.4164937819.4422237919.4679238019.4935938119.5192238219.5448238319.5703938419.5959238519.6214238619.6468838719.6723238819.6977238919.7230839019.7484239119.7737239219.7989939319.8242339419.8494339519.8746139619.8997539719.9248639819.9499439919.974984002040120.0249840220.0499440320.0748640420.0997540520.1246140620.1494440720.1742440820.1990140920.2237541020.2484641120.2731341220.2977841320.322441420.3469941520.3715541620.3960841720.4205841820.4450541920.4694942020.493942120.5182842220.5426442320.5669642420.5912642520.6155342620.6397742720.6639842820.6881642920.7123243020.7364443120.7605443220.7846143320.8086543420.8326743520.8566543620.8806143720.9045443820.9284543920.9523344020.976184412144221.023844321.0475744421.0713144521.0950244621.1187144721.1423744821.1660144921.1896245021.213245121.2367645221.2602945321.283845421.3072845521.3307345621.3541645721.3775645821.4009345921.4242946021.4476146121.4709146221.4941946321.5174346421.5406646521.5638646621.5870346721.6101846821.6333146921.6564147021.6794847121.7025347221.7255647321.7485647421.7715447521.7944947621.8174247721.8403347821.8632147921.8860748021.908948121.9317148221.954548321.977264842248522.0227248622.0454148722.0680848822.0907248922.1133449022.1359449122.1585249222.1810749322.203649422.2261149522.248649622.2710649722.293549822.3159149922.3383150022.3606850122.3830350222.4053650322.4276650422.4499450522.4722150622.4944450722.5166650822.5388650922.5610351022.5831851122.6053151222.6274251322.649551422.6715751522.6936151622.7156351722.7376351822.7596151922.7815752022.8035152122.8254252222.8473252322.8691952422.8910552522.9128852622.9346952722.9564852822.978255292353023.0217353123.0434453223.0651353323.0867953423.1084453523.1300753623.1516753723.1732653823.1948353923.2163754023.237954123.2594154223.2808954323.3023654423.3238154523.3452454623.3666454723.3880354823.409454923.4307555023.4520855123.4733955223.4946855323.5159555423.537255523.5584455623.5796555723.6008555823.6220255923.6431856023.6643256123.6854456223.7065456323.7276256423.7486856523.7697356623.7907556723.8117656823.8327556923.8537257023.8746757123.8956157223.9165257323.9374257423.958357523.979165762457724.0208257824.0416357924.0624258024.0831958124.1039458224.1246858324.1453958424.1660958524.1867758624.2074458724.2280858824.2487158924.2693259024.2899259124.3104959224.3310559324.3515959424.3721259524.3926259624.4131159724.4335859824.4540459924.4744860024.494960124.515360224.5356960324.5560660424.5764160524.5967560624.6170760724.6373760824.6576660924.6779361024.6981861124.7184161224.7386361324.7588461424.7790261524.7991961624.8193561724.8394861824.8596161924.8797162024.899862124.9198762224.9399362324.9599762424.979996252562625.0199962725.0399762825.0599362925.0798763025.099863125.1197163225.1396163325.1594963425.1793663525.1992163625.2190463725.2388663825.2586663925.2784564025.2982264125.3179864225.3377264325.3574464425.3771664525.3968564625.4165364725.4361964825.4558464925.4754865025.495165125.514765225.5342965325.5538665425.5734265525.5929765625.612565725.6320165825.6515165925.67166025.6904766125.7099266225.7293666325.7487966425.768266525.7875966625.8069866725.8263466825.845766925.8650367025.8843667125.9036767225.9229667325.9422467425.9615167525.980766762667726.0192267826.0384367926.0576368026.0768168126.0959868226.1151368326.1342768426.1533968526.172568626.191668726.2106868826.2297568926.2488169026.2678569126.2868869226.3058969326.3248969426.3438869526.3628569626.3818169726.4007669826.4196969926.4386170026.4575170126.476470226.4952870326.5141570426.53370526.5518470626.5706670726.5894770826.6082770926.6270571026.6458371126.6645871226.6833371326.7020671426.7207871526.7394871626.7581871726.7768671826.7955271926.8141872026.8328272126.8514472226.8700672326.8886672426.9072572526.9258272626.9443972726.9629472826.981487292773027.0185173127.0370173227.055573327.0739773427.0924373527.1108873627.1293273727.1477473827.1661673927.1845574027.2029474127.2213274227.2396874327.2580374427.2763674527.2946974627.31374727.331374827.3495974927.3678675027.3861375127.4043875227.4226275327.4408575427.4590675527.4772675627.4954575727.5136375827.531875927.5499576027.568176127.5862376227.6043576327.6224576427.6405576527.6586376627.6767176727.6947676827.7128176927.7308577027.7488777127.7668977227.7848977327.8028877427.8208677527.8388277627.8567877727.8747277827.8926577927.9105778027.9284878127.9463878227.9642678327.982147842878528.0178578628.0356978728.0535278828.0713478928.0891479028.1069479128.1247279228.1424979328.1602679428.1780179528.1957479628.2134779728.2311979828.2488979928.2665980028.2842780128.3019480228.319680328.3372580428.3548980528.3725280628.3901480728.4077580828.4253480928.4429381028.460581128.4780681228.4956181328.5131581428.5306981528.548281628.5657181728.5832181828.600781928.6181882028.6356482128.653182228.6705482328.6879882428.705482528.7228182628.7402282728.7576182828.7749982928.7923683028.8097283128.8270783228.8444183328.8617483428.8790683528.8963783628.9136683728.9309583828.9482383928.965584028.982758412984229.0172484329.0344684429.0516884529.0688884629.0860884729.1032684829.1204484929.137685029.1547685129.171985229.1890485329.2061685429.2232885529.2403885629.2574885729.2745685829.2916485929.308786029.3257686129.342886229.3598486329.3768686429.3938886529.4108886629.4278886729.4448686829.4618486929.4788187029.4957687129.5127187229.5296587329.5465787429.5634987529.580487629.597387729.6141987829.6310687929.6479388029.6647988129.6816488229.6984888329.7153288429.7321488529.7489588629.7657588729.7825588829.7993388929.816189029.8328789129.8496289229.8663789329.8831189429.8998389529.9165589629.9332689729.9499689829.9666589929.983339003090130.0166690230.0333190330.0499690430.0665990530.0832290630.0998390730.1164490830.1330490930.1496391030.1662191130.1827891230.1993491330.2158991430.2324391530.2489791630.2654991730.2820191830.2985191930.3150192030.331592130.3479892230.3644592330.3809292430.3973792530.4138192630.4302592730.4466792830.4630992930.479593030.495993130.5122993230.5286893330.5450593430.5614193530.5777793630.5941293730.6104693830.6267993930.6431194030.6594294130.6757294230.6920294330.7083194430.7245894530.7408594630.7571194730.7733794830.7896194930.8058495030.8220795130.8382995230.854595330.870795430.8868995530.9030795630.9192595730.9354295830.9515895930.9677396030.983879613196231.0161296331.0322496431.0483596531.0644596631.0805496731.0966296831.112796931.1287697031.1448297131.1608797231.1769197331.1929597431.2089797531.2249997631.24197731.25797831.2729997931.2889898031.3049598131.3209298231.3368898331.3528398431.3687798531.3847198631.4006498731.4165698831.4324798931.4483799031.4642799131.4801599231.4960399331.511999431.5277799531.5436299631.5594799731.5753199831.5911499931.60696100031.62278
&2016 Miniwebtool |A L G E B R A
FACTORING TRINOMIALS
F IS THE REVERSE of multiplying. &Skill in factoring, then, depends upon skill in multiplying: . &As for a
2x2 + 9x & 5
-- it will be factored as a product of binomials:
& ?)(? & ?)
The first term of each binomial will be the
of 2x2, and the
second term will be the factors of 5.
Now, how can we produce 2x2? &There is only one way: &2x&&x :
& ?)(x & ?)
And how can we produce 5? &Again, there is only one
way: &1& &5. &But does the 5 go with &2x --
(2x & 5)(x & 1)
or with& x --
(2x & 1)(x & 5) ?
Notice: &We have not yet placed any signs
How shall we decide between these two possibilities? &It is the combination that will correctly give the middle term, 9x :
2x2 + 9x & 5.
Consider the first possibility:
(2x & 5)(x & 1)
Is it possible to produce &9x& by combining the :
(that is, 2x&&1)&with& 5x ?
No, it is not. &Therefore, we must eliminate that possibility and consider the other:
(2x & 1)(x & 5)
Can we produce && by combining &10x& with 1x ?
Yes -- if we choose +5 and &1:
& 1)(x + 5)
& 1)(x + 5) = 2x2 + 9x & 5.
Skill in factoring
depends on skill in multiplying -- particularly in picking out the middle term
Problem 1.&&&Place the correct signs to give the middle term.
a) &2x2 + 7x & 15 = (2x
b) &2x2 & 7x & 15 = (2x
c) &2x2 & x & 15 = (2x
d) &2x2 & 13x + 15 = (2x
Note: &When the
is negative, as in parts a), b), c), then the
in each factor must be different. &But when the constant term is positive, as in part d), the signs must be the same.& Usually, however, that happens by itself.
Nevertheless,
can you correctly factor the following?
2x2 & 5x + 3&
= (2x & 3)(x & 1)
Problem 2.&&&Factor these trinomials.
a) &3x2 + 8x + 5&
= (3x + 5)(x + 1)
b) &3x2 + 16x + 5&
= (3x + 1)(x + 5)
c) &2x2 + 9x + 7&
= (2x + 7)(x + 1)
d) &2x2 + 15x + 7&
= (2x + 1)(x + 7)
e) &5x2 + 8x + 3&
= (5x + 3)(x + 1)
f) &5x2 + 16x + 3&
= (5x + 1)(x + 3)
Problem 3.&&& Factor these trinomials.
a) &2x2 & 7x + 5&
= (2x & 5)(x & 1)
b) &2x2 & 11x + 5&
= (2x & 1)(x & 5)
c) &3x2 + x & 10 &
= (3x & 5)(x + 2 )
d) &2x2 & x & 3 &
= (2x & 3)(x + 1)
e) &5x2 & 13x + 6&
= (5x & 3)(x & 2)
f) &5x2 & 17x + 6&
= (5x & 2)(x & 3)
g) &2x2 + 5x & 3&
= (2x & 1)(x + 3)
h) & 2x2 & 5x & 3&
= (2x + 1)(x & 3)
i) &2x2 + x & 3&
= (2x + 3)(x & 1)
j) &2x2 & 13x + 21&
= (2x & 7 )(x &3)
k) &5x2 & 7x & 6&
= (5x + 3)(x & 2)
l) &5x2 & 22x + 21&
= (5x & 7)(x & 3)
Example 1. & 1
coefficient of x2.&&&Factor &x2 + 3x & 10.
Solution.&&&The binomial factors will look like this:
(x & a)(x & b)
Since the coefficient of x2 is 1, it will not matter in which binomial we put the numbers.
(x & a)(x & b) = (x & b)(x & a).
Now, what are the factors of 10? &Let us try 2 and 5:
x2 + 3x & 10 = (x & 2)(x & 5).
We must now choose the signs so that -- as always -- the sum of the outers plus the inners will equal the middle term, which is +3x.
Choose +5 and &2.
x2 + 3x & 10 = (x & 2)(x + 5).
When the coefficient of x2 is 1, we may simply look for two numbers whose product is 10, the numerical term, and whose algebraic sum is
+3, the coefficient of&x.
&That will ensure that the sum of the outers plus the inners will equal the middle term.
the order of the factors does not matter.
(x & 2)(x + 5) = (x + 5) (x & 2).
Example 2.&&&Factor &x2 & x & 12.
& &Solution.
<TD CLASS=doc
STYLE="width: 85px"
There are several factors of 12. &Let us try 2 and 6.
But there is no way to choose signs for 6x and 2x to give the middle term, which is &x.
Let's try 3 and 4:
That will work. Choose &4 and +3.
& 4) = x2 & x & 12.
Skill in factoring depends on skill in multiplying, specifically in
constructing the middle term. (.)
Problem 4.&&&Factor. &Again, the order of the factors does not matter.
a) &x2 + 5x + 6&
= (x + 2)(x + 3)
b) &x2 & x & 6&
= (x & 3 )(x + 2)
c) &x2 + x & 6&
= (x + 3 )(x & 2)
d) &x2 & 5x + 6 &
= (x & 3)(x & 2 )
e) &x2 + 7x + 6&
(x + 1)(x + 6 )
f) &x2 & 7x + 6&
= (x & 1)(x & 6 )
g) &x2 + 5x & 6 &
= (x & 1)(x + 6 )
h) &x2 & 5x & 6 &
= (x + 1)(x & 6 )
Problem 5.&&&Factor.
a) & x2 & 10x + 9&
= (x & 1 )(x & 9)
b) &x2 + x & 12&
= (x + 4)(x & 3)
c) &x2 & 6x & 16&
= (x & 8)(x + 2)
d) &x2 & 5x & 14 &
= (x & 7)(x + 2)
e) &x2 & x & 2&
= (x + 1)(x & 2)
f) &x2 & 12x + 20&
= (x & 10 )(x & 2)
g) &x2 & 14x + 24&
= (x & 12 )(x & 2)
Example 3.&&&Factor completely
&6x8 + 30x7 + 36x6.
Solution.&&&To
factor completely means to first remove any
(Lesson 15).
6x8 + 30x7 + 36x6
6x6(x2 + 5x + 6).
&&Now continue by factoring the trinomial:
6x6(x + 2)(x + 3).
Problem 6.&&&Factor completely. &First remove any common factors.
a) &x3 + 6x2 + 5x&
= x(x2 + 6x + 5) = x(x + 5)(x + 1)
b) &x5 + 4x4 + 3x3&
= x3(x2 + 4x + 3) = x3(x + 1)(x + 3)
c) &x4 + x3 & 6x2&
= x2(x2 + x & 6) = x2(x + 3)(x & 2)
d) &4x2 & 4x & 24&
= 4(x2 & x & 6) = 4(x + 2)(x & 3)
e) &6x3 + 10x2 & 4x&
= 2x(3x2 + 5x & 2) = 2x(3x & 1)(x + 2)
f) &12x10 + 42x9 + 18x8&
= 6x8(2x2 + 7x + 3) = 6x8(2x + 1)(x + 3).
Example 4.&&&Factor by making the
&x2 + 5x & 6 = &(x2 & 5x + 6) = &(x & 2)(x & 3).
Problem 7.&&&Factor by making the leading term positive.
a) &&&x2 & 2x + 3&
= &(x2 + 2x & 3) = &(x + 3)(x & 1)
b) &&&x2 + x + 6&
= &(x2 & x & 6) = &(x +
c) &&&2x2 & 5x + 3&
= &(2x2 + 5x & 3) = &(2x & 1)(x + 3)
Quadratics in different arguments
Here is the form of a quadratic trinomial with argument x :
ax2 + bx + c.
The argument
is whatever is being squared. &x is being squared. &x is called the argument.& The argument appears in the middle term.
a, b, c are called constants. &In this quadratic,
3x2 + 2x & 1,
the constants are &3,
Now here is a quadratic whose argument is x3:
3x6 + 2x3 & 1.
x6 is the square of x3. &(Lesson 13:& .)
But that quadratic has the same constants -- 3, 2, & 1 -- as the one above. &In a sense, it is the same quadratic only with a different argument. For it is the constants that distinguish a quadratic.
Now, since the quadratic with argument x can be factored in this way:
3x2 + 2x & 1 = (3x & 1)(x + 1),
then the quadratic with argument x3is factored
the same way:
3x6 + 2x3 & 1 = (3x3 & 1)(x3 + 1).
Whenever a quadratic has constants 3, 2, &1, then for any argument, the
factoring will be
(3 times the argument & 1)(argument + 1).
Example 5.
z2 & 3z & 10
(z + 2)(z & 5).
x8 & 3x4 & 10
(x4 + 2)(x4 & 5).
The trinomials on the left have the same constants
&&1, &3, &10&& but different arguments. &That is the only difference between them.& In the first, the argument is z. &In the second, the argument is x4.
(The square of x4 is x8.)
Each quadratic is factored as
(argument + 2)(argument & 5).
Every quadratic with constants &1, &3, &10 &will be factored that way.
Problem 8.
a) &Write the form of a quadratic trinomial with argument z.
az2 + bz + c
b) &Write the form of a quadratic trinomial with argument x4.
ax8 + bx4 + c
c) &Write the form of a quadratic trinomial with argument xn.
ax2n + bxn + c
Problem 10.&&&Multiply out each of the following, which have the same constants, but different argument.
(z + 3)(z & 1)&= z2 + 2z & 3
(y + 3)(y & 1)&= y2 + 2y & 3
c) &(y6 + 3)(y6 & 1)&
= y12 + 2y6 & 3
d) &(x5 + 3)(x5 & 1)&
= x10 + 2x5 & 3
Problem 11.&&&Factor each quadratic.
a) &x2 & 6x + 5&
= (x & 1)(x & 5)
b) &z2 & 6z + 5&
= (z & 1)(z & 5)
c) &x8 & 6x4 + 5&
= (x4 & 1)(x4 & 5)
d) &x10 & 6x5 + 5&
= (x5 & 1)(x5 & 5)
e) &x6y6 & 6x3y3 + 5&
= (x3y3 & 1)(x3y3 & 5)
&f) &sin2x
(sin x & 1)(sin x
& &&&sin2x -- "sine squared x" -- means &(sin x)2.
Problem 12.&&&Factor each quadratic.
a) &x4 & x2 & 2&=
(x2 & 2)(x2 + 1)
b) &y6 + 2y3 & 8&=
(y3 + 4)(y3 & 2)
c) &z8 + 4z4 + 3&=
(z4 + 1)(z4 + 3)
d) &2x10 + 5x5 + 3&=
(2x5 + 3)(x5 + 1)
e) &x4y2 & 3x2y & 10&=
(x2y + 2)(x2y & 5)
f) &cos2x &
5 cos x + 6&=
(cos x & 3)(cox x
"cos x" is an abbreviation for the trigonometric function "cosine of angle x."
It is conventional to write the square of cos x as cos2x ("cosine squared x"). In calculus, rather than solve triangles with cos x, we do algebra with it. The above is an example. And so while you may think that in that example you are doing trigonometry, you are doing nothing but algebra.
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Lawrence Spector
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38x+23(15-x)=450 38x+345-23x=45038x-23x=450-34515x=105x=7
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