If you divide one of the roots of the equation 4x2+bx−27=0 by the other root, the quotient will be −3. find b.

12

Step-by-step explanation:

Lets use Baskara's formula to find the roots:

[-b +- sqrt (b^2 - 4ac)]/2a   =  

[-b +- sqrt (b^2 - 4*4*(-27))]/8=

[-b +- sqrt (b^2 + 432)]/8=

So,  we have 2 roots, lets call them x1 and x2. They are:

x1 =  [-b + sqrt (b^2 + 432)]/8

x2=  [-b - sqrt (b^2 + 432)]/8

Their ratio is:

x1/x2 =  [[-b + sqrt (b^2 + 432)]/8] / [[-b - sqrt (b^2 + 432)]/8] = -3

The 8s disappear

[-b + sqrt (b^2 + 432)] / [-b - sqrt (b^2 + 432)] = -3

multiplying both sides by the denominator to eliminate it and solving:

[-b + sqrt (b^2 + 432)] = -3*[-b - sqrt (b^2 + 432)]

-b + sqrt (b^2 + 432) = 3b + 3sqrt (b^2 -+ 432)

Summing b in both sides to eliminate the left b

sqrt (b^2 + 432) = 4b + 3sqrt (b^2 + 432)

Subtracting sqrt (b^2 - 432) in both sides to eliminate the left term:

0 = 4b + 3sqrt (b^2 + 432) - sqrt (b^2 + 432)

0 = 4b + 2sqrt(b^2 + 432)

Dividing both sides by 2

0 =  2b + sqrt(b^2 + 432)

Subtracting 2b in both sides:

-2b =  sqrt(b^2 + 432)

Squaring both sides and solving

4b^2 = b^2 + 432

3b^2 = 432

b^2 = 144

b = 12

Ready!

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