Question

Solve `:x^(2)-59(29)/(30)x-2=0​

Answer 1

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Solving 30x2+29x-1830 = 0 by Completing The Square .

Divide both sides of the equation by 30 to have 1 as the coefficient of the first term :

x2+(29/30)x-61 = 0

Add 61 to both side of the equation :

x2+(29/30)x = 61

Now the clever bit: Take the coefficient of x , which is 29/30 , divide by two, giving 29/60 , and finally square it giving 841/3600

Add 841/3600 to both sides of the equation :

On the right hand side we have :

61 + 841/3600 or, (61/1)+(841/3600)

The common denominator of the two fractions is 3600 Adding (219600/3600)+(841/3600) gives 220441/3600

So adding to both sides we finally get :

x2+(29/30)x+(841/3600) = 220441/3600

Adding 841/3600 has completed the left hand side into a perfect square :

x2+(29/30)x+(841/3600) =

(x+(29/60)) • (x+(29/60)) =

(x+(29/60))2

Things which are equal to the same thing are also equal to one another.

Solving 30x2+29x-1830 = 0 by the Quadratic Formula .

According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :

- B ± √ B2-4AC

x = ————————

2A

In our case, A = 30

B = 29

C = -1830

Accordingly, B2 - 4AC =

841 - (-219600) =

220441

Applying the quadratic formula :

-29 ± √ 220441

x = ————————

60

√ 220441 , rounded to 4 decimal digits, is 469.5114

So now we are looking at:

x = ( -29 ± 469.511 ) / 60

Two real solutions:

x =(-29+√220441)/60= 7.342

or:

x =(-29-√220441)/60=-8.309

Two solutions were found :

x =(-29-√220441)/60=-8.309

x =(-29+√220441)/60= 7.342

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