Solve quadratic equation:-
39x^2 - 17x -9=0
Answers
Answer:
(17 ± √1693)/78
Step-by-step explanation:
Given Quadratic Equation is 39x² - 17x - 9 = 0
On comparing with ax² + bx + c = 0, we get
a = 39, b = -17, c = -9.
∴ D = b² - 4ac
= (-17)² - 4(39)(-9)
= 1693
The solutions are:
x = -b ± √D/2a
= -(-17) ± √1693/78
= (17 ± √1693)/78
Therefore, the roots of the quadratic equation are:
∴ (17 + √1693)/78, (17 - √1693)/78
Hope it helps!
Solving 39x2-17x-9 = 0 by Completing The Square .
Divide both sides of the equation by 39 to have 1 as the coefficient of the first term :
x2-(17/39)x-(3/13) = 0
Add 3/13 to both side of the equation :
x2-(17/39)x = 3/13
Now the clever bit: Take the coefficient of x , which is 17/39 , divide by two, giving 17/78 , and finally square it giving 289/6084
Add 289/6084 to both sides of the equation :
On the right hand side we have :
3/13 + 289/6084 The common denominator of the two fractions is 6084 Adding (1404/6084)+(289/6084) gives 1693/6084
So adding to both sides we finally get :
x2-(17/39)x+(289/6084) = 1693/6084
Adding 289/6084 has completed the left hand side into a perfect square :
x2-(17/39)x+(289/6084) =
(x-(17/78)) • (x-(17/78)) =
(x-(17/78))2
Things which are equal to the same thing are also equal to one another. Since
x2-(17/39)x+(289/6084) = 1693/6084 and
x2-(17/39)x+(289/6084) = (x-(17/78))2
then, according to the law of transitivity,
(x-(17/78))2 = 1693/6084
We'll refer to this Equation as Eq. #3.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(17/78))2 is
(x-(17/78))2/2 =
(x-(17/78))1 =
x-(17/78)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x-(17/78) = √ 1693/6084
Add 17/78 to both sides to obtain:
x = 17/78 + √ 1693/6084
Since a square root has two values, one positive and the other negative
x2 - (17/39)x - (3/13) = 0
has two solutions:
x = 17/78 + √ 1693/6084
or
x = 17/78 - √ 1693/6084
Note that √ 1693/6084 can be written as
√ 1693 / √ 6084 which is √ 1693 / 78