SOLUTION: Solve the equation by factoring. 25x squared + 30x + 8 = 0

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Question 1172888: Solve the equation by factoring.
25x squared + 30x + 8 = 0

Found 3 solutions by Boreal, ikleyn, greenestamps:
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
25x^2+30x+8=0
divide by 25 first and multiply the 8 by 25
x^2+30x+200=0
now factor that if possible (x+20)(x+10)=0
divide the constant by the 25 multiplied by earlier and reduce to lowest terms
(x+(4/5))(x+(2/5))=0
and put the denominator out front.
(5x+4)(5x+2)=0
x=-4/5, -2/5
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.

Introduce new variable  u = 5x.

Then the equation takes the form


    u^2 + 6u + 8 = 0.


Factor it mentally in 1 second


    (u + 4)*(u + 2) = 0


The roots are  u= -4  and  u= -2.


Hence, the roots of the original equations are


    a)  5x = -4,  x = ,

and

    b)  5x = -2,  x = .

Solved // mentally.



Answer by greenestamps(13198)   (Show Source): You can put this solution on YOUR website!


Two other tutors have shown methods for factoring quadratics like this one using different purely algebraic methods.

Both methods are good; and there are many other similar methods. But I prefer the basic method that -- at least when I was learning algebra 60 years ago -- was always used.

We want to factor the expression as



where a, b, c, and d are integers.

The positive constant term 8 tells us the constants b and d are the same sign; then the positive linear term 30x tells us b and d are both positive.

a and c have to multiply to 25; and b and d have to multiply to 8. There are not many possibilities:
a and c are 25 and 1, or 5 and 5
b and d are 1 and 8, or 2 and 4, or 4 and 2, or 8 and 1

Try different combinations until you find the one that gives the correct middle term, 30x.

Fast and easy calculations show the answer to be
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