find the roots of 4xsquare+ 3x+5=0 by the method of completing the square.
Answers
Step-by-step explanation:
Hope these will help you
hi mate,
solution :
Solving 4x²+3x+5 = 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 = 4
B = 3
C = 5
Accordingly, B² - 4AC = 9 - 80 = -71
Applying the quadratic formula :
-3 ± √ -71
x = ——————
8
In the set of real numbers, negative numbers do not have square roots. A new set of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)
Both i and -i are the square roots of minus 1
Accordingly,√ -71 =
√ 71 • (-1) =
√ 71 • √ -1 =
± √ 71 • i
√ 71 , rounded to 4 decimal digits, is 8.4261
So now we are looking at:
x = ( -3 ± 8.426 i ) / 8
Two imaginary solutions :
x =(-3+√-71)/8 = (-3+i√ 71 )/8 =
-0.3750+1.0533i
or:
x =(-3-√-71)/8=(-3-i√ 71 )/8=
-0.3750-1.0533i
Two solutions were found :
x = (-3-√-71)/8=(-3-i√ 71 )/8
= - 0.3750-1.0533i
x =(-3+√-71)/8=(-3+i√71 )8
= - 0.3750+1.0533i
Two solutions were found :
x =(-3-√-71)/8=(-3-i√ 71 )/8
= -0.3750-1.0533i
x =(-3+√-71)/8=(-3+i√ 71 )/8
= -0.3750+1.0533i
or..
Solving 4x2+3x+5 = 0 by Completing The Square .
Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x2+(3/4)x+(5/4) = 0
Subtract 5/4 from both side of the equation :
x2+(3/4)x = -5/4
Now the clever bit: Take the coefficient of x , which is 3/4 , divide by two, giving 3/8 , and finally square it giving 9/64
Add 9/64 to both sides of the equation :
On the right hand side we have :
-5/4 + 9/64 The common denominator of the two fractions is 64 Adding (-80/64)+(9/64) gives -71/64
So adding to both sides we finally get :
x2+(3/4)x+(9/64) = -71/64
Adding 9/64 has completed the left hand side into a perfect square :
x2+(3/4)x+(9/64) =
(x+(3/8)) • (x+(3/8)) =
(x+(3/8))2
Things which are equal to the same thing are also equal to one another. Since
x2+(3/4)x+(9/64) = -71/64 and
x2+(3/4)x+(9/64) = (x+(3/8))2
then, according to the law of transitivity,
(x+(3/8))2 = -71/64
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+(3/8))2 is
(x+(3/8))2/2 =
(x+(3/8))1 =
x+(3/8)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x+(3/8) = √ -71/64
Subtract 3/8 from both sides to obtain:
x = -3/8 + √ -71/64
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Since a square root has two values, one positive and the other negative
x2 + (3/4)x + (5/4) = 0
has two solutions:
x = -3/8 + √ 71/64 • i
or
x = -3/8 - √ 71/64 • i
Note that √ 71/64 can be written as
√ 71 / √ 64 which is √ 71 / 8
i hope it helps you.