I am having a difficult time understanding what I am doing wrong with quadratics.
$5 x^{2} - 15 x - 4 = 0$

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I am having a difficult time understanding what I am doing wrong with quadratics.5x2−15x−4=0

German Ferguson · Accepted Answer

Let us follow your approach: till (5x+5)(x−4)=−16, all your statements are correct. Now you still need to find which values of x satisfy this, and those are the roots.However none of the values you mention satisfy - for e.g. taking x=−12, this statement reads(−60+5)(−12−4)=−55⋅−16=880≠−16.If the RHS of -16 factors as a×b for some a,b, and you can find a solution for x which simultaneously satisfies 5x+5=a,x−4=b, indeed you have a solution. However such factors aren&#039;t obvious and in general finding them may be tougher than solving the original quadratic through other methods. Further, you may get only one value of x through this approach, for the other root, you will have to keep checking other possible factorisations.The only case where you can independently set any one the factors in LHS to equal the RHS to find roots, is when the RHS is 0, for obvious reasons. This is what you seem to have done, but except when RHS is 0, this may not solve the original problem. Another e.g. where you may find this working is (2x−3)(3−x)=1, due to the coincidence of the factorisation 12=1, even so that gives you only one of the roots.Hence this approach isn&#039;t useful in most cases, and others have suggested valid approaches such as completing the square, or using the quadratic formula, both of which works always.

Malia Booth · Accepted Answer

Are you trying to complete the square in your problem?5x2−15x−4=0Then you&#039;d need to do this:5x2−15x=4Then divide by 5:x2−3x=45Now complete the square:x2−3x+(−32)2=45+(−32)2(x−32)2=45+94(x−32)2=6120(x−32)2=(6120)2(x−32)2−(6120)2=0Now you can factor this using a2−b2=(a−b)(a+b)(x−32−6120)(x−32+6120)=0Now you can use that ab=0 means a=0 or b=0.So x−32−6120=0or x−32+6120=0So solving these two givesx=32+12615≈3.2464or x=32−12615≈0.2464I could also have done it like this:5x2−15x=45(x2−3x)=45(x2−3x+(−32)2)−5(−32)2=45(x−32)2−454=45(x−32)2=614(x−32)2=6120Then take the square root of both sides. That givesx−32=6120or x−32=−6120which gives the same two solutions.

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